 4.57: 7. Find a basis for the vector space of all diagonalmatrices. What ...
 4.4.173: Linear Combinations In Exercises 14, write eachvector as a linear c...
 4.4.454: Determining Solutions of a Differential Equation InExercises 112, d...
 4.4.246: Writing the Standard Basis In Exercises 16, writethe standard basis...
 4.4.538: Vector Operations In Exercises 14, find (a)(b) (c) and (d) 3u2v.
 4.4.324: Row Vectors and Column Vectors In Exercises 14,write (a) the row ve...
 4.4.66: Describing the Additive Identity In Exercises 16,describe the zero ...
 4.4.648: One solution of the homogeneous linear system is and Explain whyand...
 4.4.653: Consider the following subspaces of V R3. Find and W Z.
 4.4.114: Verifying Subspaces In Exercises 16, verify thatis a subspace V of ...
 4.4.400: Finding a Coordinate Matrix In Exercises 14,find the coordinate mat...
 4.4.1: Finding the Component Form of a Vector InExercises 1 and 2, find th...
 4.4.174: Linear Combinations In Exercises 14, write eachvector as a linear c...
 4.4.455: Determining Solutions of a Differential Equation InExercises 112, d...
 4.4.247: Writing the Standard Basis In Exercises 16, writethe standard basis...
 4.4.539: Vector Operations In Exercises 14, find (a)(b) (c) and (d) 3u2v.
 4.4.325: Row Vectors and Column Vectors In Exercises 14,write (a) the row ve...
 4.4.67: Describing the Additive Identity In Exercises 16,describe the zero ...
 4.4.649: The vectors and are solutions of the homogeneous linear systemExpla...
 4.4.654: If and are subspaces of such that and thenprove that every vector i...
 4.4.115: Verifying Subspaces In Exercises 16, verify thatis a subspace V of ...
 4.4.401: Finding a Coordinate Matrix In Exercises 14,find the coordinate mat...
 4.4.2: Finding the Component Form of a Vector InExercises 1 and 2, find th...
 4.4.175: Linear Combinations In Exercises 14, write eachvector as a linear c...
 4.4.456: Determining Solutions of a Differential Equation InExercises 112, d...
 4.4.248: Writing the Standard Basis In Exercises 16, writethe standard basis...
 4.4.540: Vector Operations In Exercises 14, find (a)(b) (c) and (d) 3u2v.
 4.4.326: Row Vectors and Column Vectors In Exercises 14,write (a) the row ve...
 4.4.68: Describing the Additive Identity In Exercises 16,describe the zero ...
 4.4.650: Consider the two systems represented by the augmented matrices. If ...
 4.4.655: Let and suppose that is a basis for the subspaceand is a basis for ...
 4.4.116: Verifying Subspaces In Exercises 16, verify thatis a subspace V of ...
 4.4.402: Finding a Coordinate Matrix In Exercises 14,find the coordinate mat...
 4.4.3: Representing a Vector In Exercises 36, use a directedline segment t...
 4.4.176: Linear Combinations In Exercises 14, write eachvector as a linear c...
 4.4.457: Determining Solutions of a Differential Equation InExercises 112, d...
 4.4.249: Writing the Standard Basis In Exercises 16, writethe standard basis...
 4.4.541: Vector Operations In Exercises 14, find (a)(b) (c) and (d) 3u2v.
 4.4.327: Row Vectors and Column Vectors In Exercises 14,write (a) the row ve...
 4.4.69: Describing the Additive Identity In Exercises 16,describe the zero ...
 4.4.651: The vectors and are solutions of the linear system Is the vectorals...
 4.4.656: Consider the subspaces andof Show that Is the direct sum of and Wha...
 4.4.117: Verifying Subspaces In Exercises 16, verify thatis a subspace V of ...
 4.4.403: Finding a Coordinate Matrix In Exercises 14,find the coordinate mat...
 4.4.4: Representing a Vector In Exercises 36, use a directedline segment t...
 4.4.177: In Exercises 58, for the matricesandin M2,2, determine whether the ...
 4.4.458: Determining Solutions of a Differential Equation InExercises 112, d...
 4.4.250: Writing the Standard Basis In Exercises 16, writethe standard basis...
 4.4.542: Solving a Vector Equation In Exercises 58,solve for where u 1, 1, 2...
 4.4.328: Finding a Basis for a Row Space and Rank InExercises 510, find (a) ...
 4.4.70: Describing the Additive Identity In Exercises 16,describe the zero ...
 4.4.652: The vectors and are solutions of the linear system Is the vectorals...
 4.4.657: Do there exist two twodimensional subspaces of R3 whose intersecti...
 4.4.118: Verifying Subspaces In Exercises 16, verify thatis a subspace V of ...
 4.4.404: Finding a Coordinate Matrix In Exercises 510, giventhe coordinate m...
 4.4.5: Representing a Vector In Exercises 36, use a directedline segment t...
 4.4.178: In Exercises 58, for the matricesandin M2,2, determine whether the ...
 4.4.459: Determining Solutions of a Differential Equation InExercises 112, d...
 4.4.251: Writing the Standard Basis In Exercises 16, writethe standard basis...
 4.4.543: Solving a Vector Equation In Exercises 58,solve for where u 1, 1, 2...
 4.4.329: Finding a Basis for a Row Space and Rank InExercises 510, find (a) ...
 4.4.71: Describing the Additive Identity In Exercises 16,describe the zero ...
 4.4.119: Verifying Subspaces In Exercises 16, verify thatis a subspace V of ...
 4.4.405: Finding a Coordinate Matrix In Exercises 510, giventhe coordinate m...
 4.4.6: Representing a Vector In Exercises 36, use a directedline segment t...
 4.4.179: In Exercises 58, for the matricesandin M2,2, determine whether the ...
 4.4.460: Determining Solutions of a Differential Equation InExercises 112, d...
 4.4.252: Writing In Exercises 714, explain why is not a basis for R2
 4.4.544: Solving a Vector Equation In Exercises 58,solve for where u 1, 1, 2...
 4.4.330: Finding a Basis for a Row Space and Rank InExercises 510, find (a) ...
 4.4.72: Describing the Additive Inverse In Exercises 712,describe the addit...
 4.4.120: Subsets That Are Not Subspaces In Exercises 720, Wis not a subspace...
 4.4.406: Finding a Coordinate Matrix In Exercises 510, giventhe coordinate m...
 4.4.7: Finding the Sum of Two Vectors In Exercises 710, findthe sum of the...
 4.4.180: In Exercises 58, for the matricesandin M2,2, determine whether the ...
 4.4.461: Determining Solutions of a Differential Equation InExercises 112, d...
 4.4.253: Writing In Exercises 714, explain why is not a basis for R3
 4.4.545: Solving a Vector Equation In Exercises 58,solve for where u 1, 1, 2...
 4.4.331: Finding a Basis for a Row Space and Rank InExercises 510, find (a) ...
 4.4.73: Describing the Additive Inverse In Exercises 712,describe the addit...
 4.4.121: Subsets That Are Not Subspaces In Exercises 720, Wis not a subspace...
 4.4.407: Finding a Coordinate Matrix In Exercises 510, giventhe coordinate m...
 4.4.8: Finding the Sum of Two Vectors In Exercises 710, findthe sum of the...
 4.4.181: Spanning Sets In Exercises 920, determine whetherthe set spans R2. ...
 4.4.462: Determining Solutions of a Differential Equation InExercises 112, d...
 4.4.254: Writing In Exercises 714, explain why is not a basis for R4
 4.4.546: Writing a Linear Combination In Exercises 912, writeas a linear com...
 4.4.332: Finding a Basis for a Row Space and Rank InExercises 510, find (a) ...
 4.4.74: Describing the Additive Inverse In Exercises 712,describe the addit...
 4.4.122: Subsets That Are Not Subspaces In Exercises 720, Wis not a subspace...
 4.4.408: Finding a Coordinate Matrix In Exercises 510, giventhe coordinate m...
 4.4.9: Finding the Sum of Two Vectors In Exercises 710, findthe sum of the...
 4.4.182: Spanning Sets In Exercises 920, determine whetherthe set spans R2. ...
 4.4.463: Determining Solutions of a Differential Equation InExercises 112, d...
 4.4.255: Writing In Exercises 714, explain why is not a basis for R5
 4.4.547: Writing a Linear Combination In Exercises 912, writeas a linear com...
 4.4.333: Finding a Basis for a Row Space and Rank InExercises 510, find (a) ...
 4.4.75: Describing the Additive Inverse In Exercises 712,describe the addit...
 4.4.123: Subsets That Are Not Subspaces In Exercises 720, Wis not a subspace...
 4.4.409: Finding a Coordinate Matrix In Exercises 510, giventhe coordinate m...
 4.4.10: Finding the Sum of Two Vectors In Exercises 710, findthe sum of the...
 4.4.183: Spanning Sets In Exercises 920, determine whetherthe set spans R2. ...
 4.4.464: Determining Solutions of a Differential Equation InExercises 112, d...
 4.4.256: Writing In Exercises 714, explain why is not a basis for R6
 4.4.548: Writing a Linear Combination In Exercises 912, writeas a linear com...
 4.4.334: Finding a Basis for a Subspace In Exercises 1114,find a basis for t...
 4.4.76: Describing the Additive Inverse In Exercises 712,describe the addit...
 4.4.124: Subsets That Are Not Subspaces In Exercises 720, Wis not a subspace...
 4.4.410: Finding a Coordinate Matrix In Exercises 1116,find the coordinate m...
 4.4.11: Vector Operations In Exercises 1116, find thevector and illustrate ...
 4.4.184: Spanning Sets In Exercises 920, determine whetherthe set spans R2. ...
 4.4.465: Determining Solutions of a Differential Equation InExercises 112, d...
 4.4.257: Writing In Exercises 714, explain why is not a basis for R7
 4.4.549: Writing a Linear Combination In Exercises 912, writeas a linear com...
 4.4.335: Finding a Basis for a Subspace In Exercises 1114,find a basis for t...
 4.4.77: Describing the Additive Inverse In Exercises 712,describe the addit...
 4.4.125: Subsets That Are Not Subspaces In Exercises 720, Wis not a subspace...
 4.4.411: Finding a Coordinate Matrix In Exercises 1116,find the coordinate m...
 4.4.12: Vector Operations In Exercises 1116, find thevector and illustrate ...
 4.4.185: Spanning Sets In Exercises 920, determine whetherthe set spans R2. ...
 4.4.466: Finding the Wronskian for a Set of Functions InExercises 1326, find...
 4.4.258: Writing In Exercises 714, explain why is not a basis for R8
 4.4.550: Describing the Zero Vector and the Additive InverseIn Exercises 131...
 4.4.336: Finding a Basis for a Subspace In Exercises 1114,find a basis for t...
 4.4.78: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.126: Subsets That Are Not Subspaces In Exercises 720, Wis not a subspace...
 4.4.412: Finding a Coordinate Matrix In Exercises 1116,find the coordinate m...
 4.4.13: Vector Operations In Exercises 1116, find thevector and illustrate ...
 4.4.186: Spanning Sets In Exercises 920, determine whetherthe set spans R2. ...
 4.4.467: Finding the Wronskian for a Set of Functions InExercises 1326, find...
 4.4.259: Writing In Exercises 714, explain why is not a basis for R9
 4.4.551: Describing the Zero Vector and the Additive InverseIn Exercises 131...
 4.4.337: Finding a Basis for a Subspace In Exercises 1114,find a basis for t...
 4.4.79: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.127: Subsets That Are Not Subspaces In Exercises 720, Wis not a subspace...
 4.4.413: Finding a Coordinate Matrix In Exercises 1116,find the coordinate m...
 4.4.14: Vector Operations In Exercises 1116, find thevector and illustrate ...
 4.4.187: Spanning Sets In Exercises 920, determine whetherthe set spans R2. ...
 4.4.468: Finding the Wronskian for a Set of Functions InExercises 1326, find...
 4.4.260: Writing In Exercises 1520, explain why is not abasis for R3
 4.4.552: Describing the Zero Vector and the Additive InverseIn Exercises 131...
 4.4.338: Finding a Basis for a Subspace In Exercises 1518,find a basis for t...
 4.4.80: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.128: Subsets That Are Not Subspaces In Exercises 720, Wis not a subspace...
 4.4.414: Finding a Coordinate Matrix In Exercises 1116,find the coordinate m...
 4.4.15: Vector Operations In Exercises 1116, find thevector and illustrate ...
 4.4.188: Spanning Sets In Exercises 920, determine whetherthe set spans R2. ...
 4.4.469: Finding the Wronskian for a Set of Functions InExercises 1326, find...
 4.4.261: Writing In Exercises 1520, explain why is not abasis for R3
 4.4.553: Describing the Zero Vector and the Additive InverseIn Exercises 131...
 4.4.339: Finding a Basis for a Subspace In Exercises 1518,find a basis for t...
 4.4.81: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.129: Subsets That Are Not Subspaces In Exercises 720, Wis not a subspace...
 4.4.415: Finding a Coordinate Matrix In Exercises 1116,find the coordinate m...
 4.4.16: Vector Operations In Exercises 1116, find thevector and illustrate ...
 4.4.189: Spanning Sets In Exercises 920, determine whetherthe set spans R2. ...
 4.4.470: Finding the Wronskian for a Set of Functions InExercises 1326, find...
 4.4.262: Writing In Exercises 1520, explain why is not abasis for R3
 4.4.554: Determining Subspaces In Exercises 1724,determine whether is a subs...
 4.4.340: Finding a Basis for a Subspace In Exercises 1518,find a basis for t...
 4.4.82: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.130: Subsets That Are Not Subspaces In Exercises 720, Wis not a subspace...
 4.4.416: Finding a Transition Matrix In Exercises 1724, findthe transition m...
 4.4.17: Given the vector v 2, 1, sketch (a) (b) and (c) 12v.
 4.4.190: Spanning Sets In Exercises 920, determine whetherthe set spans R2. ...
 4.4.471: Finding the Wronskian for a Set of Functions InExercises 1326, find...
 4.4.263: Writing In Exercises 1520, explain why is not abasis for R3
 4.4.555: Determining Subspaces In Exercises 1724,determine whether is a subs...
 4.4.341: Finding a Basis for a Subspace In Exercises 1518,find a basis for t...
 4.4.83: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.131: Subsets That Are Not Subspaces In Exercises 720, Wis not a subspace...
 4.4.417: Finding a Transition Matrix In Exercises 1724, findthe transition m...
 4.4.18: Given the vector sketch (a) (b)and (c) 0v.
 4.4.191: Spanning Sets In Exercises 920, determine whetherthe set spans R2. ...
 4.4.472: Finding the Wronskian for a Set of Functions InExercises 1326, find...
 4.4.264: Writing In Exercises 1520, explain why is not abasis for R3
 4.4.556: Determining Subspaces In Exercises 1724,determine whether is a subs...
 4.4.342: Finding a Basis for a Column Space and Rank InExercises 1924, find ...
 4.4.84: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.132: Subsets That Are Not Subspaces In Exercises 720, Wis not a subspace...
 4.4.418: Finding a Transition Matrix In Exercises 1724, findthe transition m...
 4.4.19: Vector Operations In Exercises 1924, let u 1, 2, 3, v 2, 2, 1, w 4,...
 4.4.192: Spanning Sets In Exercises 920, determine whetherthe set spans R2. ...
 4.4.473: Finding the Wronskian for a Set of Functions InExercises 1326, find...
 4.4.265: Writing In Exercises 1520, explain why is not abasis for R3
 4.4.557: Determining Subspaces In Exercises 1724,determine whether is a subs...
 4.4.343: Finding a Basis for a Column Space and Rank InExercises 1924, find ...
 4.4.85: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.133: Subsets That Are Not Subspaces In Exercises 720, Wis not a subspace...
 4.4.419: Finding a Transition Matrix In Exercises 1724, findthe transition m...
 4.4.20: Vector Operations In Exercises 1924, let u 1, 2, 3, v 2, 2, 1, w 4,...
 4.4.193: Spanning Sets In Exercises 2126, determine whetherthe set spans R3 ...
 4.4.474: Finding the Wronskian for a Set of Functions InExercises 1326, find...
 4.4.266: Writing In Exercises 2124, explain why is not abasis for P2
 4.4.558: Determining Subspaces In Exercises 1724,determine whether is a subs...
 4.4.344: Finding a Basis for a Column Space and Rank InExercises 1924, find ...
 4.4.86: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.134: Determining Subspaces In Exercises 2128,determine whether the subse...
 4.4.420: Finding a Transition Matrix In Exercises 1724, findthe transition m...
 4.4.21: Vector Operations In Exercises 1924, let u 1, 2, 3, v 2, 2, 1, w 4,...
 4.4.194: Spanning Sets In Exercises 2126, determine whetherthe set spans R3 ...
 4.4.475: Finding the Wronskian for a Set of Functions InExercises 1326, find...
 4.4.267: Writing In Exercises 2124, explain why is not abasis for P2
 4.4.559: Determining Subspaces In Exercises 1724,determine whether is a subs...
 4.4.345: Finding a Basis for a Column Space and Rank InExercises 1924, find ...
 4.4.87: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.135: Determining Subspaces In Exercises 2128,determine whether the subse...
 4.4.421: Finding a Transition Matrix In Exercises 1724, findthe transition m...
 4.4.22: Vector Operations In Exercises 1924, let u 1, 2, 3, v 2, 2, 1, w 4,...
 4.4.195: Spanning Sets In Exercises 2126, determine whetherthe set spans R3 ...
 4.4.476: Finding the Wronskian for a Set of Functions InExercises 1326, find...
 4.4.268: Writing In Exercises 2124, explain why is not abasis for P2
 4.4.560: Determining Subspaces In Exercises 1724,determine whether is a subs...
 4.4.346: Finding a Basis for a Column Space and Rank InExercises 1924, find ...
 4.4.88: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.136: Determining Subspaces In Exercises 2128,determine whether the subse...
 4.4.422: Finding a Transition Matrix In Exercises 1724, findthe transition m...
 4.4.23: Vector Operations In Exercises 1924, let u 1, 2, 3, v 2, 2, 1, w 4,...
 4.4.24: Vector Operations In Exercises 1924, let u 1, 2, 3, v 2, 2, 1, w 4,...
 4.4.196: Spanning Sets In Exercises 2126, determine whetherthe set spans R3 ...
 4.4.477: Finding the Wronskian for a Set of Functions InExercises 1326, find...
 4.4.269: Writing In Exercises 2124, explain why is not abasis for P2
 4.4.561: Determining Subspaces In Exercises 1724,determine whether is a subs...
 4.4.347: Finding a Basis for a Column Space and Rank InExercises 1924, find ...
 4.4.89: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.137: Determining Subspaces In Exercises 2128,determine whether the subse...
 4.4.423: Finding a Transition Matrix In Exercises 1724, findthe transition m...
 4.4.25: Given the vector v 1, 2, 2, sketch (a) v, (b) 2v, and (c) 2vand v.
 4.4.197: Spanning Sets In Exercises 2126, determine whetherthe set spans R3 ...
 4.4.478: Finding the Wronskian for a Set of Functions InExercises 1326, find...
 4.4.270: Writing In Exercises 2528, explain why is not abasis for M2,2.
 4.4.562: Which of the subsets of is a subspace of
 4.4.348: Finding the Nullspace of a Matrix In Exercises 2536,find the nullsp...
 4.4.90: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.138: Determining Subspaces In Exercises 2128,determine whether the subse...
 4.4.424: Finding a Transition Matrix In Exercises 2534,use a software progra...
 4.4.26: Given the vector sketch (a) (b)and (c) v.
 4.4.198: Spanning Sets In Exercises 2126, determine whetherthe set spans R3 ...
 4.4.479: Finding the Wronskian for a Set of Functions InExercises 1326, find...
 4.4.271: Writing In Exercises 2528, explain why is not abasis for M2,2.
 4.4.563: Which of the subsets of is a subspace of R3?
 4.4.349: Finding the Nullspace of a Matrix In Exercises 2536,find the nullsp...
 4.4.91: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.139: Determining Subspaces In Exercises 2128,determine whether the subse...
 4.4.425: Finding a Transition Matrix In Exercises 2534,use a software progra...
 4.4.27: Which of the following vectors are scalar multiples of 3, 2, 5?
 4.4.199: Determine whether the set S 1, x2, x2 2spans P2
 4.4.480: Testing for Linear Independence In Exercises 2734,(a) verify that e...
 4.4.272: Writing In Exercises 2528, explain why is not abasis for M2,2.
 4.4.564: Spanning Sets, Linear Independence, and Bases InExercises 2732, det...
 4.4.350: Finding the Nullspace of a Matrix In Exercises 2536,find the nullsp...
 4.4.92: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.140: Determining Subspaces In Exercises 2128,determine whether the subse...
 4.4.426: Finding a Transition Matrix In Exercises 2534,use a software progra...
 4.4.28: Which of the following vectors are scalar multiples of z 12, 23, 34?
 4.4.200: Determine whether the set S x2 2x, x3 8, x3 x2, x2 4 spans p3
 4.4.481: Testing for Linear Independence In Exercises 2734,(a) verify that e...
 4.4.273: Writing In Exercises 2528, explain why is not abasis for M2,2.
 4.4.565: Spanning Sets, Linear Independence, and Bases InExercises 2732, det...
 4.4.351: Finding the Nullspace of a Matrix In Exercises 2536,find the nullsp...
 4.4.93: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.141: Determining Subspaces In Exercises 2128,determine whether the subse...
 4.4.427: Finding a Transition Matrix In Exercises 2534,use a software progra...
 4.4.29: In Exercises 29 and 30, find (a) u v, (b) 2u 3v and (c) 2v u.
 4.4.201: Testing for Linear Independence In Exercises 2940,determine whether...
 4.4.482: Testing for Linear Independence In Exercises 2734,(a) verify that e...
 4.4.274: Determining Whether a Set Is a Basis In Exercises2932, determine wh...
 4.4.566: Spanning Sets, Linear Independence, and Bases InExercises 2732, det...
 4.4.352: Finding the Nullspace of a Matrix In Exercises 2536,find the nullsp...
 4.4.94: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.142: Determining Subspaces In Exercises 2936,determine whether the subse...
 4.4.428: Finding a Transition Matrix In Exercises 2534,use a software progra...
 4.4.30: In Exercises 29 and 30, find (a) u v, (b) 2u 3v and (c) 2v u.
 4.4.202: Testing for Linear Independence In Exercises 2940,determine whether...
 4.4.483: Testing for Linear Independence In Exercises 2734,(a) verify that e...
 4.4.275: Determining Whether a Set Is a Basis In Exercises2932, determine wh...
 4.4.567: Spanning Sets, Linear Independence, and Bases InExercises 2732, det...
 4.4.353: Finding the Nullspace of a Matrix In Exercises 2536,find the nullsp...
 4.4.95: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.143: Determining Subspaces In Exercises 2936,determine whether the subse...
 4.4.429: Finding a Transition Matrix In Exercises 2534,use a software progra...
 4.4.31: Vector Operations In Exercises 31 and 32, usea graphing utility wit...
 4.4.203: Testing for Linear Independence In Exercises 2940,determine whether...
 4.4.484: Testing for Linear Independence In Exercises 2734,(a) verify that e...
 4.4.276: Determining Whether a Set Is a Basis In Exercises2932, determine wh...
 4.4.568: Spanning Sets, Linear Independence, and Bases InExercises 2732, det...
 4.4.354: Finding the Nullspace of a Matrix In Exercises 2536,find the nullsp...
 4.4.96: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.144: Determining Subspaces In Exercises 2936,determine whether the subse...
 4.4.430: Finding a Transition Matrix In Exercises 2534,use a software progra...
 4.4.32: Vector Operations In Exercises 31 and 32, usea graphing utility wit...
 4.4.204: Testing for Linear Independence In Exercises 2940,determine whether...
 4.4.485: Testing for Linear Independence In Exercises 2734,(a) verify that e...
 4.4.277: Determining Whether a Set Is a Basis In Exercises2932, determine wh...
 4.4.569: Spanning Sets, Linear Independence, and Bases InExercises 2732, det...
 4.4.355: Finding the Nullspace of a Matrix In Exercises 2536,find the nullsp...
 4.4.97: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.145: Determining Subspaces In Exercises 2936,determine whether the subse...
 4.4.431: Finding a Transition Matrix In Exercises 2534,use a software progra...
 4.4.33: Solving a Vector Equation In Exercises 3336, solvefor where u 1, 1,...
 4.4.205: Testing for Linear Independence In Exercises 2940,determine whether...
 4.4.486: Testing for Linear Independence In Exercises 2734,(a) verify that e...
 4.4.278: Determining Whether a Set Is a Basis In Exercises3340, determine wh...
 4.4.570: Determine whether S 1 t, 2t 3t 2, t 2 2t 3, 2 t 3S is a basis for P3.
 4.4.356: Finding the Nullspace of a Matrix In Exercises 2536,find the nullsp...
 4.4.98: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.146: Determining Subspaces In Exercises 2936,determine whether the subse...
 4.4.432: Finding a Transition Matrix In Exercises 2534,use a software progra...
 4.4.34: Solving a Vector Equation In Exercises 3336, solvefor where u 1, 1,...
 4.4.206: Testing for Linear Independence In Exercises 2940,determine whether...
 4.4.487: Testing for Linear Independence In Exercises 2734,(a) verify that e...
 4.4.279: Determining Whether a Set Is a Basis In Exercises3340, determine wh...
 4.4.571: Determine whether S 1 t, 2t 3t 2, t 2 2t 3, 2 t 3 is a basis for S P2.
 4.4.357: Finding the Nullspace of a Matrix In Exercises 2536,find the nullsp...
 4.4.99: Testing for a Vector Space In Exercises 1334,determine whether the ...
 4.4.147: Determining Subspaces In Exercises 2936,determine whether the subse...
 4.4.433: Finding a Transition Matrix In Exercises 2534,use a software progra...
 4.4.434: Finding Transition and Coordinate MatricesIn Exercises 3538, (a) fi...
 4.4.35: Solving a Vector Equation In Exercises 3336, solvefor where u 1, 1,...
 4.4.207: Testing for Linear Independence In Exercises 2940,determine whether...
 4.4.488: Pendulum Consider a pendulum of length thatswings by the force of g...
 4.4.280: Determining Whether a Set Is a Basis In Exercises3340, determine wh...
 4.4.572: Determine whether is a basis forDetermining Whether a Set Is a Basi...
 4.4.358: Finding the Nullspace of a Matrix In Exercises 2536,find the nullsp...
 4.4.100: Rather than use the standard definitions of additionand scalar mult...
 4.4.148: Determining Subspaces In Exercises 2936,determine whether the subse...
 4.4.435: Finding Transition and Coordinate MatricesIn Exercises 3538, (a) fi...
 4.4.36: Solving a Vector Equation In Exercises 3336, solvefor where u 1, 1,...
 4.4.208: Testing for Linear Independence In Exercises 2940,determine whether...
 4.4.489: Proof Prove that is the general solution of
 4.4.281: Determining Whether a Set Is a Basis In Exercises3340, determine wh...
 4.4.573: Determine whether is a basis forDetermining Whether a Set Is a Basi...
 4.4.359: Finding the Nullspace of a Matrix In Exercises 2536,find the nullsp...
 4.4.101: Rather than use the standard definitions of additionand scalar mult...
 4.4.149: Determining Subspaces In Exercises 2936,determine whether the subse...
 4.4.436: Finding Transition and Coordinate MatricesIn Exercises 3538, (a) fi...
 4.4.37: Solving a Vector Equation In Exercises 37 and 38,find such that 2u ...
 4.4.209: Testing for Linear Independence In Exercises 2940,determine whether...
 4.4.490: Proof In Exercises 3739, prove that the given setof solutions of a ...
 4.4.282: Determining Whether a Set Is a Basis In Exercises3340, determine wh...
 4.4.574: Finding the Nullspace, Nullity, and Rank of a MatrixIn Exercises 37...
 4.4.360: Finding a Basis and Dimension In Exercises 3746,find (a) a basis fo...
 4.4.102: Proof Prove in full detail that with the standardoperations, is a v...
 4.4.150: Determining Subspaces In Exercises 3742,determine whether the set W...
 4.4.437: Finding Transition and Coordinate MatricesIn Exercises 3538, (a) fi...
 4.4.38: Solving a Vector Equation In Exercises 37 and 38,find such that 2u ...
 4.4.210: Testing for Linear Independence In Exercises 2940,determine whether...
 4.4.491: Proof In Exercises 3739, prove that the given setof solutions of a ...
 4.4.283: Determining Whether a Set Is a Basis In Exercises3340, determine wh...
 4.4.575: Finding the Nullspace, Nullity, and Rank of a MatrixIn Exercises 37...
 4.4.361: Finding a Basis and Dimension In Exercises 3746,find (a) a basis fo...
 4.4.103: Proof Prove in full detail that the set is areal number with the st...
 4.4.151: Determining Subspaces In Exercises 3742,determine whether the set W...
 4.4.438: Finding Transition and Coordinate MatricesIn Exercises 3941, use a ...
 4.4.39: Writing a Linear Combination In Exercises 3944,write as a linear co...
 4.4.211: Testing for Linear Independence In Exercises 2940,determine whether...
 4.4.492: Proof In Exercises 3739, prove that the given setof solutions of a ...
 4.4.284: Determining Whether a Set Is a Basis In Exercises3340, determine wh...
 4.4.576: Finding the Nullspace, Nullity, and Rank of a MatrixIn Exercises 37...
 4.4.362: Finding a Basis and Dimension In Exercises 3746,find (a) a basis fo...
 4.4.104: Determine whether the set with the operations x1, y1 x2, y2 x1x2, y...
 4.4.152: Determining Subspaces In Exercises 3742,determine whether the set W...
 4.4.439: Finding Transition and Coordinate MatricesIn Exercises 3941, use a ...
 4.4.40: Writing a Linear Combination In Exercises 3944,write as a linear co...
 4.4.212: Testing for Linear Independence In Exercises 2940,determine whether...
 4.4.493: Proof Let be a set of solutions of a secondorderlinear homogeneous...
 4.4.285: Determining Whether a Set Is a Basis In Exercises3340, determine wh...
 4.4.577: Finding the Nullspace, Nullity, and Rank of a MatrixIn Exercises 37...
 4.4.363: Finding a Basis and Dimension In Exercises 3746,find (a) a basis fo...
 4.4.105: (a) Describe the conditions under which a set may beclassified as a...
 4.4.153: Determining Subspaces In Exercises 3742,determine whether the set W...
 4.4.440: Finding Transition and Coordinate MatricesIn Exercises 3941, use a ...
 4.4.41: Writing a Linear Combination In Exercises 3944,write as a linear co...
 4.4.213: Testing for Linear Independence In Exercises 4144,determine whether...
 4.4.494: Writing Is the sum of two solutions of anonhomogeneous linear diffe...
 4.4.286: Determining Whether a Set Is a Basis In Exercises4144, determine wh...
 4.4.578: Finding the Nullspace, Nullity, and Rank of a MatrixIn Exercises 37...
 4.4.364: Finding a Basis and Dimension In Exercises 3746,find (a) a basis fo...
 4.4.106: MassSpring System The mass in a massspringsystem (see figure) is ...
 4.4.154: Determining Subspaces In Exercises 3742,determine whether the set W...
 4.4.441: Let and be two bases for(a) When write the transition matrix from t...
 4.4.42: Writing a Linear Combination In Exercises 3944,write as a linear co...
 4.4.214: Testing for Linear Independence In Exercises 4144,determine whether...
 4.4.495: Writing Is the scalar multiple of a solution ofa nonhomogeneous lin...
 4.4.287: Determining Whether a Set Is a Basis In Exercises4144, determine wh...
 4.4.579: Finding the Nullspace, Nullity, and Rank of a MatrixIn Exercises 37...
 4.4.365: Finding a Basis and Dimension In Exercises 3746,find (a) a basis fo...
 4.4.107: Let be the set of all infinite sequences of realnumbers, with the o...
 4.4.155: Determining Subspaces In Exercises 3742,determine whether the set W...
 4.4.442: Coordinate Representation in In Exercises 4346,find the coordinate ...
 4.4.43: Writing a Linear Combination In Exercises 3944,write as a linear co...
 4.4.215: Testing for Linear Independence In Exercises 4144,determine whether...
 4.4.496: Identifying and Graphing a Conic Section InExercises 4360, identify...
 4.4.288: Determining Whether a Set Is a Basis In Exercises4144, determine wh...
 4.4.580: Finding a Basis and Dimension In Exercises 4346,find (a) a basis fo...
 4.4.366: Finding a Basis and Dimension In Exercises 3746,find (a) a basis fo...
 4.4.108: Let be the set of all positive real numbers. Determinewhether is a ...
 4.4.156: True or False? In Exercises 43 and 44, determinewhether each statem...
 4.4.443: Coordinate Representation in In Exercises 4346,find the coordinate ...
 4.4.44: Writing a Linear Combination In Exercises 3944,write as a linear co...
 4.4.216: Testing for Linear Independence In Exercises 4144,determine whether...
 4.4.497: Identifying and Graphing a Conic Section InExercises 4360, identify...
 4.4.289: Determining Whether a Set Is a Basis In Exercises4144, determine wh...
 4.4.581: Finding a Basis and Dimension In Exercises 4346,find (a) a basis fo...
 4.4.367: Finding a Basis and Dimension In Exercises 3746,find (a) a basis fo...
 4.4.109: Proof Complete the proof of the cancellation propertyof vector addi...
 4.4.157: (a) Every vector space contains two proper subspacesthat are the ze...
 4.4.444: Coordinate Representation in In Exercises 4346,find the coordinate ...
 4.4.45: Writing a Linear Combination In Exercises 4548,write as a linear co...
 4.4.217: Testing for Linear Independence In Exercises 4548,determine whether...
 4.4.498: Identifying and Graphing a Conic Section InExercises 4360, identify...
 4.4.290: Determining Whether a Set Is a Basis In Exercises 45and 46, determi...
 4.4.582: Finding a Basis and Dimension In Exercises 4346,find (a) a basis fo...
 4.4.368: Finding a Basis and Dimension In Exercises 3746,find (a) a basis fo...
 4.4.110: True or False? In Exercises 45 and 46, determinewhether each statem...
 4.4.158: Consider the vector spaceswhere is the set of all polynomials of de...
 4.4.445: Coordinate Representation in In Exercises 4346,find the coordinate ...
 4.4.46: Writing a Linear Combination In Exercises 4548,write as a linear co...
 4.4.218: Testing for Linear Independence In Exercises 4548,determine whether...
 4.4.499: Identifying and Graphing a Conic Section InExercises 4360, identify...
 4.4.291: Determining Whether a Set Is a Basis In Exercises 45and 46, determi...
 4.4.583: Finding a Basis and Dimension In Exercises 4346,find (a) a basis fo...
 4.4.369: Finding a Basis and Dimension In Exercises 3746,find (a) a basis fo...
 4.4.111: (a) To show that a set is not a vector space, it issufficient to sh...
 4.4.159: Calculus Let and be defined as inExample 5. Show that is a subspace...
 4.4.446: Coordinate Representation in In Exercises4750, find the coordinate ...
 4.4.47: Writing a Linear Combination In Exercises 4548,write as a linear co...
 4.4.219: Testing for Linear Independence In Exercises 4548,determine whether...
 4.4.500: Identifying and Graphing a Conic Section InExercises 4360, identify...
 4.4.292: Determining Whether a Set Is a Basis In Exercises4750, determine wh...
 4.4.584: Finding a Basis for a Row Space and RankIn Exercises 4752, find (a)...
 4.4.370: Rank, Nullity, Bases, and Linear IndependenceIn Exercises 47 and 48...
 4.4.112: Proof Prove Property 1 of Theorem 4.4.
 4.4.160: Calculus Let be the vector space of realvaluedfunctions defined on ...
 4.4.447: Coordinate Representation in In Exercises4750, find the coordinate ...
 4.4.48: Writing a Linear Combination In Exercises 4548,write as a linear co...
 4.4.220: Testing for Linear Independence In Exercises 4548,determine whether...
 4.4.501: Identifying and Graphing a Conic Section InExercises 4360, identify...
 4.4.293: Determining Whether a Set Is a Basis In Exercises4750, determine wh...
 4.4.585: Finding a Basis for a Row Space and RankIn Exercises 4752, find (a)...
 4.4.371: Rank, Nullity, Bases, and Linear IndependenceIn Exercises 47 and 48...
 4.4.113: Proof Prove Property 4 of Theorem 4.4.
 4.4.161: Calculus Determine whether the setis a subspace C0, 1. of Prove you...
 4.4.162: Let be the subset of consisting of all points on aline that passes ...
 4.4.448: Coordinate Representation in In Exercises4750, find the coordinate ...
 4.4.49: Writing a Linear Combination In Exercises 49 and 50,use a software ...
 4.4.221: Showing Linear Dependence In Exercises 4952, showthat the set is li...
 4.4.502: Identifying and Graphing a Conic Section InExercises 4360, identify...
 4.4.294: Determining Whether a Set Is a Basis In Exercises4750, determine wh...
 4.4.586: Finding a Basis for a Row Space and RankIn Exercises 4752, find (a)...
 4.4.372: Nonhomogeneous System In Exercises 4954,(a) determine whether the n...
 4.4.163: Explain why it is sufficientto test for closure in order to establi...
 4.4.449: Coordinate Representation in In Exercises4750, find the coordinate ...
 4.4.50: Writing a Linear Combination In Exercises 49 and 50,use a software ...
 4.4.222: Showing Linear Dependence In Exercises 4952, showthat the set is li...
 4.4.503: Identifying and Graphing a Conic Section InExercises 4360, identify...
 4.4.295: Determining Whether a Set Is a Basis In Exercises4750, determine wh...
 4.4.587: Finding a Basis for a Row Space and RankIn Exercises 4752, find (a)...
 4.4.373: Nonhomogeneous System In Exercises 4954,(a) determine whether the n...
 4.4.164: Guided Proof Prove that a nonempty set is asubspace of a vector spa...
 4.4.450: True or False? In Exercises 51 and 52, determinewhether each statem...
 4.4.51: True or False? In Exercises 51 and 52, determinewhether each statem...
 4.4.223: Showing Linear Dependence In Exercises 4952, showthat the set is li...
 4.4.504: Identifying and Graphing a Conic Section InExercises 4360, identify...
 4.4.296: Determining the Dimension of a Vector Space InExercises 5156, deter...
 4.4.588: Finding a Basis for a Row Space and RankIn Exercises 4752, find (a)...
 4.4.374: Nonhomogeneous System In Exercises 4954,(a) determine whether the n...
 4.4.165: Let and be vectors in a vector space Show thatthe set of all linear...
 4.4.451: True or False? In Exercises 51 and 52, determinewhether each statem...
 4.4.52: (a) To subtract two vectors in subtract theircorresponding componen...
 4.4.224: Showing Linear Dependence In Exercises 4952, showthat the set is li...
 4.4.505: Identifying and Graphing a Conic Section InExercises 4360, identify...
 4.4.297: Determining the Dimension of a Vector Space InExercises 5156, deter...
 4.4.589: Finding a Basis for a Row Space and RankIn Exercises 4752, find (a)...
 4.4.375: Nonhomogeneous System In Exercises 4954,(a) determine whether the n...
 4.4.166: Proof Let be a fixed matrix. Prove that the set W x R3: Ax 12
 4.4.452: Let be the transition matrix from to and letbe the transition matri...
 4.4.53: Writing a Linear Combination In Exercises 53 and 54,the zero vector...
 4.4.225: For which values of is each set linearly independent?
 4.4.506: Identifying and Graphing a Conic Section InExercises 4360, identify...
 4.4.298: Determining the Dimension of a Vector Space InExercises 5156, deter...
 4.4.590: Finding a Coordinate Matrix In Exercises 5964, findthe coordinate m...
 4.4.376: Nonhomogeneous System In Exercises 4954,(a) determine whether the n...
 4.4.167: Proof Let be a fixed matrix. Prove that the set W x Rn : Ax 0 is a ...
 4.4.453: Let be the transition matrix from to and letbe the transition matri...
 4.4.54: Writing a Linear Combination In Exercises 53 and 54,the zero vector...
 4.4.226: For which values of is each set linearly independent?
 4.4.507: Identifying and Graphing a Conic Section InExercises 4360, identify...
 4.4.299: Determining the Dimension of a Vector Space InExercises 5156, deter...
 4.4.591: Finding a Coordinate Matrix In Exercises 5964, findthe coordinate m...
 4.4.377: Nonhomogeneous System In Exercises 4954,(a) determine whether the n...
 4.4.168: Proof Let be a subspace of the vector space Provethat the zero vect...
 4.4.55: Illustrate properties 110 of Theorem 4.2 for u2, 1, 3, 6, v 1, 4, 0...
 4.4.227: Proof Complete the proof of Theorem 4.7.
 4.4.508: Identifying and Graphing a Conic Section InExercises 4360, identify...
 4.4.300: Determining the Dimension of a Vector Space InExercises 5156, deter...
 4.4.592: Finding a Coordinate Matrix In Exercises 5964, findthe coordinate m...
 4.4.378: Consistency of In Exercises 5558, determinewhether is in the column...
 4.4.169: Give an example showing that the union of twosubspaces of a vector ...
 4.4.56: Illustrate properties 110 of Theorem 4.2 for uu 2, 1, 3, v 3, 4, 0,...
 4.4.228: By inspection, determine whyeach of the sets is linearly dependent.
 4.4.509: Identifying and Graphing a Conic Section InExercises 4360, identify...
 4.4.301: Determining the Dimension of a Vector Space InExercises 5156, deter...
 4.4.593: Finding a Coordinate Matrix In Exercises 5964, findthe coordinate m...
 4.4.379: Consistency of In Exercises 5558, determinewhether is in the column...
 4.4.170: Proof Let and be fixed matrices. Prove thatthe set W X : XAB BAX is...
 4.4.57: Proof Complete the proof of Theorem 4.1.
 4.4.229: Spanning the Same Subspace In Exercises 57 and 58,show that the set...
 4.4.510: Identifying and Graphing a Conic Section InExercises 4360, identify...
 4.4.594: Finding a Coordinate Matrix In Exercises 5964, findthe coordinate m...
 4.4.380: Consistency of In Exercises 5558, determinewhether is in the column...
 4.4.171: Proof Let and be two subspaces of a vectorspace(a) Prove that the s...
 4.4.58: Proof Prove each property of vector addition andscalar multiplicati...
 4.4.230: Spanning the Same Subspace In Exercises 57 and 58,show that the set...
 4.4.511: Identifying and Graphing a Conic Section InExercises 4360, identify...
 4.4.303: Find a basis for the vector space of all symmetricmatrices. What is...
 4.4.595: Finding a Coordinate Matrix In Exercises 5964, findthe coordinate m...
 4.4.381: Consistency of In Exercises 5558, determinewhether is in the column...
 4.4.172: 59. Proof Complete the proof of Theorem 4.6 by showingthat the inte...
 4.4.59: Writing Let be a system of linear equationsin variables. Designate ...
 4.4.231: True or False? In Exercises 59 and 60, determinewhether each statem...
 4.4.512: Identifying and Graphing a Conic Section InExercises 4360, identify...
 4.4.304: Find all subsets of the set S 1, 0, 0, 1, 1, 13 3 that form a basis...
 4.4.596: Finding a Coordinate Matrix In Exercises 5964, findthe coordinate m...
 4.4.382: Proof Prove that if is not square, then either the rowvectors of or...
 4.4.60: Consider the vectorsand(a) Use directed line segments to represent ...
 4.4.232: True or False? In Exercises 59 and 60, determinewhether each statem...
 4.4.513: Identifying and Graphing a Conic Section InExercises 4360, identify...
 4.4.305: Find all subsets of the set S 1, 3, 2, 4, 1, 1, 2, 7, 3, 2, 1, 1R2....
 4.4.597: Finding a Coordinate Matrix In Exercises 5964, findthe coordinate m...
 4.4.383: Give an example showing that the rank of the product oftwo matrices...
 4.4.61: Proof In Exercises 6164, complete the proofs of theremaining proper...
 4.4.233: In Exercises 61 and 62, prove that the set ofvectors is linearly in...
 4.4.514: Matching an Equation with a Graph In Exercises6164, match the graph...
 4.4.306: Find a basis for that includes the vector 2, 2. 2
 4.4.598: Finding a Coordinate Matrix In Exercises 5964, findthe coordinate m...
 4.4.384: Give examples of matrices and of the same sizesuch that
 4.4.62: Proof In Exercises 6164, complete the proofs of theremaining proper...
 4.4.234: In Exercises 61 and 62, prove that the set ofvectors is linearly in...
 4.4.515: Matching an Equation with a Graph In Exercises6164, match the graph...
 4.4.307: Find a basis for that includes the vectors s and 0, 1, 1.R
 4.4.599: Finding a Coordinate Matrix In Exercises 5964, findthe coordinate m...
 4.4.385: Proof Prove that the nonzero row vectors of a matrixin rowechelon ...
 4.4.63: Proof In Exercises 6164, complete the proofs of theremaining proper...
 4.4.235: Guided Proof Prove that a nonempty subset of afinite set of linearl...
 4.4.516: Matching an Equation with a Graph In Exercises6164, match the graph...
 4.4.308: Geometric Description, Basis, and Dimension InExercises 63 and 64, ...
 4.4.600: Finding a Coordinate Matrix In Exercises 5964, findthe coordinate m...
 4.4.64: Proof In Exercises 6164, complete the proofs of theremaining proper...
 4.4.387: Show that the three points andin a plane are collinear if and only ...
 4.4.236: Proof Prove that if is a nonempty subset of thefinite set and is li...
 4.4.517: Matching an Equation with a Graph In Exercises6164, match the graph...
 4.4.309: Geometric Description, Basis, and Dimension InExercises 63 and 64, ...
 4.4.601: Finding a Coordinate Matrix In Exercises 5964, findthe coordinate m...
 4.4.65: Writing How could you describe vector subtractiongeometrically? Wha...
 4.4.237: Proof Prove that any set of vectors containing thezero vector is li...
 4.4.518: Rotation of a Conic Section In Exercises 6576,perform a rotation of...
 4.4.310: Geometric Description, Basis, and Dimension InExercises 65 and 66, ...
 4.4.602: Finding a Transition Matrix In Exercises 6568, findthe transition m...
 4.4.388: Given matrices and show that the row vectors ofare in the row space...
 4.4.238: Proof When is a linearlyindependent set of vectors and the setis li...
 4.4.519: Rotation of a Conic Section In Exercises 6576,perform a rotation of...
 4.4.311: Geometric Description, Basis, and Dimension InExercises 65 and 66, ...
 4.4.603: Finding a Transition Matrix In Exercises 6568, findthe transition m...
 4.4.389: Find the ranks of the matrix for and 4. Can you find a pattern in t...
 4.4.239: Proof Let be a linearly independentset of vectors in a vector space...
 4.4.520: Rotation of a Conic Section In Exercises 6576,perform a rotation of...
 4.4.312: Basis and Dimension In Exercises 6770, find (a) abasis for and (b) ...
 4.4.604: Finding a Transition Matrix In Exercises 6568, findthe transition m...
 4.4.390: Proof Prove each property of the system of linearequations in varia...
 4.4.240: Proof Let be a linearly independentset of vectors in a vector space...
 4.4.521: Rotation of a Conic Section In Exercises 6576,perform a rotation of...
 4.4.313: Basis and Dimension In Exercises 6770, find (a) abasis for and (b) ...
 4.4.605: Finding a Transition Matrix In Exercises 6568, findthe transition m...
 4.4.391: Proof Let be an matrix. Prove that NA NATA.
 4.4.241: Proof Let be a linearly independent set.Prove that the set is linea...
 4.4.522: Rotation of a Conic Section In Exercises 6576,perform a rotation of...
 4.4.314: Basis and Dimension In Exercises 6770, find (a) abasis for and (b) ...
 4.4.606: Finding Transition and Coordinate MatricesIn Exercises 6972, (a) fi...
 4.4.392: True or False? In Exercises 6971, determine whethereach statement i...
 4.4.242: Let and be any three vectors from a vectorspace Determine whether t...
 4.4.523: Rotation of a Conic Section In Exercises 6576,perform a rotation of...
 4.4.315: Basis and Dimension In Exercises 6770, find (a) abasis for and (b) ...
 4.4.607: Finding Transition and Coordinate MatricesIn Exercises 6972, (a) fi...
 4.4.393: (a) If an matrix is rowequivalent to anmatrix then the row space o...
 4.4.243: Proof Let be a nonsingular matrix of order 3. Provethat if is a lin...
 4.4.524: Rotation of a Conic Section In Exercises 6576,perform a rotation of...
 4.4.316: True or False? In Exercises 71 and 72, determinewhether each statem...
 4.4.608: Finding Transition and Coordinate MatricesIn Exercises 6972, (a) fi...
 4.4.394: (a) If an matrix can be obtained fromelementary row operations on a...
 4.4.244: Writing Under what conditions will a set consistingof a single vect...
 4.4.525: Rotation of a Conic Section In Exercises 6576,perform a rotation of...
 4.4.317: True or False? In Exercises 71 and 72, determinewhether each statem...
 4.4.609: Finding Transition and Coordinate MatricesIn Exercises 6972, (a) fi...
 4.4.395: . The dimension of the row spaceof a matrix is 2.(a) What is the di...
 4.4.245: Proof Prove the corollary to Theorem 4.8: Twovectors and are linear...
 4.4.526: Rotation of a Conic Section In Exercises 6576,perform a rotation of...
 4.4.318: Proof Prove that if is a basisfor a vector space and is a nonzero s...
 4.4.610: Let be the subspace of [the set of all polynomialsof degree 3 or le...
 4.4.396: Let and be square matrices of order satisfyingfor all in(a) Find th...
 4.4.319: Proof Prove Theorem 4.12.
 4.4.527: Rotation of a Conic Section In Exercises 6576,perform a rotation of...
 4.4.611: Calculus Let the vector space ofall continuously differentiable fun...
 4.4.397: . Proof Let be an matrix.(a) Prove that the system of linear equati...
 4.4.320: Proof Prove that if is a subspace of a finitedimensional vector spa...
 4.4.528: Rotation of a Conic Section In Exercises 6576,perform a rotation of...
 4.4.612: Writing Let B p1x, p2x, . . . , pnx, pn1xVbe a basis for Must conta...
 4.4.398: . Proof Prove that row operations do not change thedependency relat...
 4.4.321: (a) A set consists of two vectors of the formExplain why is not a b...
 4.4.529: Rotation of a Conic Section In Exercises 6576,perform a rotation of...
 4.4.613: Proof Let and be matrices withand . Prove that if is symmetric and ...
 4.4.399: Writing Explain why the row vectors of a 4 x 3matrix form a linearl...
 4.4.322: Proof Let be a linearly independent set of vectorsfrom the finite d...
 4.4.530: Rotation of a Degenerate Conic Section In Exercises7780, perform a ...
 4.4.614: Proof Let and consider the set of allpolynomials of the form where ...
 4.4.323: . Guided Proof Let be a spanning set for the finitedimensional vect...
 4.4.531: Rotation of a Degenerate Conic Section In Exercises7780, perform a ...
 4.4.615: 8. Let and be three linearly independentvectors in a vector space I...
 4.4.532: Rotation of a Degenerate Conic Section In Exercises7780, perform a ...
 4.4.616: Proof Let be an square matrix. Prove that therow vectors of are lin...
 4.4.533: Rotation of a Degenerate Conic Section In Exercises7780, perform a ...
 4.4.617: . Proof Let be an square matrix, and let be ascalar. Prove that the...
 4.4.534: . Proof Prove that a rotation of will eliminatethe term from the e...
 4.4.618: Let and(a) Show that and are linearly independent in(b) Show that a...
 4.4.535: . Proof Prove that a rotation of where cot 2 a cb, xy will eliminat...
 4.4.619: Given a set of functions, describe how its domain caninfluence whet...
 4.4.536: Proof For the equation definethe matrix as A ab2b2c .A (a) Prove th...
 4.4.620: True or False? In Exercises 8386, determine whethereach statement i...
 4.4.537: (a) Prove that if then the graph ofis a line.(b) Prove that if then...
 4.4.621: True or False? In Exercises 8386, determine whethereach statement i...
 4.4.622: True or False? In Exercises 8386, determine whethereach statement i...
 4.4.623: True or False? In Exercises 8386, determine whethereach statement i...
 4.4.624: Determining Solutions of a Differential EquationIn Exercises 8790, ...
 4.4.625: Determining Solutions of a Differential EquationIn Exercises 8790, ...
 4.4.626: Determining Solutions of a Differential EquationIn Exercises 8790, ...
 4.4.627: Determining Solutions of a Differential EquationIn Exercises 8790, ...
 4.4.628: Finding the Wronskian for a Set of FunctionsIn Exercises 9194, find...
 4.4.629: Finding the Wronskian for a Set of FunctionsIn Exercises 9194, find...
 4.4.630: Finding the Wronskian for a Set of FunctionsIn Exercises 9194, find...
 4.4.631: Finding the Wronskian for a Set of FunctionsIn Exercises 9194, find...
 4.4.632: Identifying and Graphing a Conic SectionIn Exercises 99106, identif...
 4.4.633: Identifying and Graphing a Conic SectionIn Exercises 99106, identif...
 4.4.634: Identifying and Graphing a Conic SectionIn Exercises 99106, identif...
 4.4.635: Identifying and Graphing a Conic SectionIn Exercises 99106, identif...
 4.4.636: Identifying and Graphing a Conic SectionIn Exercises 99106, identif...
 4.4.637: Identifying and Graphing a Conic SectionIn Exercises 99106, identif...
 4.4.638: Identifying and Graphing a Conic SectionIn Exercises 99106, identif...
 4.4.639: Identifying and Graphing a Conic SectionIn Exercises 99106, identif...
 4.4.640: Identifying and Graphing a Conic SectionIn Exercises 99106, identif...
 4.4.641: Identifying and Graphing a Conic SectionIn Exercises 99106, identif...
 4.4.642: Identifying and Graphing a Conic SectionIn Exercises 99106, identif...
 4.4.643: Identifying and Graphing a Conic SectionIn Exercises 99106, identif...
 4.4.644: Rotation of a Conic Section In Exercises 107110,perform a rotation ...
 4.4.645: Rotation of a Conic Section In Exercises 107110,perform a rotation ...
 4.4.646: Rotation of a Conic Section In Exercises 107110,perform a rotation ...
 4.4.647: Rotation of a Conic Section In Exercises 107110,perform a rotation ...
Solutions for Chapter 4: Vector Spaces
Full solutions for Elementary Linear Algebra  7th Edition
ISBN: 9781133110873
Solutions for Chapter 4: Vector Spaces
Get Full SolutionsElementary Linear Algebra was written by and is associated to the ISBN: 9781133110873. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4: Vector Spaces includes 656 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 7. Since 656 problems in chapter 4: Vector Spaces have been answered, more than 10957 students have viewed full stepbystep solutions from this chapter.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.