 6.6.316: Finding an Image and a Preimage In Exercises 14,find (a) the image ...
 6.6.85: Finding the Kernel of a Linear TransformationIn Exercises 110, find...
 6.6.414: Find the standard matrix L for for the line x = 0.
 6.6.420: . Find the standard matrix for the projection onto the axis. That ...
 6.6.152: The Standard Matrix for a Linear TransformationIn Exercises 16, fin...
 6.6.1: Finding an Image and a Preimage In Exercises 18,use the function to...
 6.6.210: Finding a Matrix for a Linear Transformation InExercises 18, find t...
 6.6.248: Let be a reflection in the axis. Find theimage of each vector.
 6.6.317: Finding an Image and a Preimage In Exercises 14,find (a) the image ...
 6.6.86: Finding the Kernel of a Linear TransformationIn Exercises 110, find...
 6.6.415: Find the standard matrix L for for the line y = 0.
 6.6.421: Find the standard matrix for the projection onto the axis.
 6.6.153: The Standard Matrix for a Linear TransformationIn Exercises 16, fin...
 6.6.2: Finding an Image and a Preimage In Exercises 18,use the function to...
 6.6.211: Finding a Matrix for a Linear Transformation InExercises 18, find t...
 6.6.249: Let be a reflection in the axis. Find theimage of each vector.
 6.6.318: Finding an Image and a Preimage In Exercises 14,find (a) the image ...
 6.6.87: Finding the Kernel of a Linear TransformationIn Exercises 110, find...
 6.6.416: Find the standard matrix L for for the line x  y 0=L
 6.6.422: Consider the line represented by Find a vector parallel to andanoth...
 6.6.154: The Standard Matrix for a Linear TransformationIn Exercises 16, fin...
 6.6.3: Finding an Image and a Preimage In Exercises 18,use the function to...
 6.6.212: Finding a Matrix for a Linear Transformation InExercises 18, find t...
 6.6.250: Let be a reflection in the line Find theimage of each vector.
 6.6.319: Finding an Image and a Preimage In Exercises 14,find (a) the image ...
 6.6.88: Finding the Kernel of a Linear TransformationIn Exercises 110, find...
 6.6.417: . Consider the line represented by Find a vector parallel to andano...
 6.6.423: . Consider the general line represented by Find a vectorparallel to...
 6.6.155: The Standard Matrix for a Linear TransformationIn Exercises 16, fin...
 6.6.4: Finding an Image and a Preimage In Exercises 18,use the function to...
 6.6.213: Finding a Matrix for a Linear Transformation InExercises 18, find t...
 6.6.251: Let be a reflection in the line Findthe image of each vector
 6.6.320: Linear Transformations and Standard Matrices InExercises 512, deter...
 6.6.89: Finding the Kernel of a Linear TransformationIn Exercises 110, find...
 6.6.418: Consider the general line represented by Find a vectorparallel to a...
 6.6.424: Use Figure 6.24 to show that where is the reflection inthe line Sol...
 6.6.156: The Standard Matrix for a Linear TransformationIn Exercises 16, fin...
 6.6.5: Finding an Image and a Preimage In Exercises 18,use the function to...
 6.6.214: Finding a Matrix for a Linear Transformation InExercises 18, find t...
 6.6.252: Let and(a) Determine for any(b) Give a geometric description of T.
 6.6.321: Linear Transformations and Standard Matrices InExercises 512, deter...
 6.6.90: Finding the Kernel of a Linear TransformationIn Exercises 110, find...
 6.6.419: Find the standard matrix for the reflection in the line Use thismat...
 6.6.157: The Standard Matrix for a Linear TransformationIn Exercises 16, fin...
 6.6.6: Finding an Image and a Preimage In Exercises 18,use the function to...
 6.6.215: Finding a Matrix for a Linear Transformation InExercises 18, find t...
 6.6.253: Let and(a) Determine for any(b) Give a geometric description of T.
 6.6.322: Linear Transformations and Standard Matrices InExercises 512, deter...
 6.6.91: Finding the Kernel of a Linear TransformationIn Exercises 110, find...
 6.6.158: Finding the Image of a Vector In Exercises 710, usethe standard mat...
 6.6.7: Finding an Image and a Preimage In Exercises 18,use the function to...
 6.6.216: Finding a Matrix for a Linear Transformation InExercises 18, find t...
 6.6.254: Identifying and Representing a Transformation InExercises 714, (a) ...
 6.6.323: Linear Transformations and Standard Matrices InExercises 512, deter...
 6.6.92: Finding the Kernel of a Linear TransformationIn Exercises 110, find...
 6.6.159: Finding the Image of a Vector In Exercises 710, usethe standard mat...
 6.6.8: Finding an Image and a Preimage In Exercises 18,use the function to...
 6.6.217: Finding a Matrix for a Linear Transformation InExercises 18, find t...
 6.6.255: Identifying and Representing a Transformation InExercises 714, (a) ...
 6.6.324: Linear Transformations and Standard Matrices InExercises 512, deter...
 6.6.93: Finding the Kernel of a Linear TransformationIn Exercises 110, find...
 6.6.160: Finding the Image of a Vector In Exercises 710, usethe standard mat...
 6.6.9: Linear Transformations In Exercises 922, determinewhether the funct...
 6.6.218: Let andbe bases for and letbe the matrix for relative to(a) Find th...
 6.6.256: Identifying and Representing a Transformation InExercises 714, (a) ...
 6.6.325: Linear Transformations and Standard Matrices InExercises 512, deter...
 6.6.94: Finding the Kernel of a Linear TransformationIn Exercises 110, find...
 6.6.161: Finding the Image of a Vector In Exercises 710, usethe standard mat...
 6.6.10: Linear Transformations In Exercises 922, determinewhether the funct...
 6.6.219: Repeat Exercise 9 forand(Use matrix given in Exercise 9.)
 6.6.257: Identifying and Representing a Transformation InExercises 714, (a) ...
 6.6.326: Linear Transformations and Standard Matrices InExercises 512, deter...
 6.6.95: Finding Bases for the Kernel and Range InExercises 1118, represents...
 6.6.162: Finding the Standard Matrix and the Image InExercises 1122, (a) fin...
 6.6.11: Linear Transformations In Exercises 922, determinewhether the funct...
 6.6.220: Let andbe bases for and letbe the matrix for relative to(a) Find th...
 6.6.258: Identifying and Representing a Transformation InExercises 714, (a) ...
 6.6.327: Linear Transformations and Standard Matrices InExercises 512, deter...
 6.6.96: Finding Bases for the Kernel and Range InExercises 1118, represents...
 6.6.163: Finding the Standard Matrix and the Image InExercises 1122, (a) fin...
 6.6.12: Linear Transformations In Exercises 922, determinewhether the funct...
 6.6.221: . Repeat Exercise 11 for ,and(Use matrix given in Exercise 11.)
 6.6.259: Identifying and Representing a Transformation InExercises 714, (a) ...
 6.6.328: Let be a linear transformation from into suchthat and Findand T0, 1.T
 6.6.97: Finding Bases for the Kernel and Range InExercises 1118, represents...
 6.6.164: Finding the Standard Matrix and the Image InExercises 1122, (a) fin...
 6.6.13: Linear Transformations In Exercises 922, determinewhether the funct...
 6.6.222: Let andbe bases for and letbe the matrix for relative to(a) Find th...
 6.6.260: Identifying and Representing a Transformation InExercises 714, (a) ...
 6.6.329: . Let be a linear transformation from into suchthat andFind T0, 1, ...
 6.6.98: Finding Bases for the Kernel and Range InExercises 1118, represents...
 6.6.165: Finding the Standard Matrix and the Image InExercises 1122, (a) fin...
 6.6.14: Linear Transformations In Exercises 922, determinewhether the funct...
 6.6.223: Repeat Exercise 13 forand(Use matrix given in Exercise 13.)
 6.6.261: Identifying and Representing a Transformation InExercises 714, (a) ...
 6.6.330: Let be a linear transformation from into suchthat and Find T0, 1.T1,
 6.6.99: Finding Bases for the Kernel and Range InExercises 1118, represents...
 6.6.166: Finding the Standard Matrix and the Image InExercises 1122, (a) fin...
 6.6.15: Linear Transformations In Exercises 922, determinewhether the funct...
 6.6.224: Similar Matrices In Exercises 1518, use the matrixto show that the ...
 6.6.262: Finding Fixed Points of a Linear Transformation InExercises 1522, f...
 6.6.263: Finding Fixed Points of a Linear Transformation InExercises 1522, f...
 6.6.331: . Let be a linear transformation from into suchthat and Find T2, 4.T
 6.6.100: Finding Bases for the Kernel and Range InExercises 1118, represents...
 6.6.167: Finding the Standard Matrix and the Image InExercises 1122, (a) fin...
 6.6.16: Linear Transformations In Exercises 922, determinewhether the funct...
 6.6.225: Similar Matrices In Exercises 1518, use the matrixto show that the ...
 6.6.264: Finding Fixed Points of a Linear Transformation InExercises 1522, f...
 6.6.332: Linear Transformation Given by a Matrix In Exercises1724, define th...
 6.6.101: Finding Bases for the Kernel and Range InExercises 1118, represents...
 6.6.168: Finding the Standard Matrix and the Image InExercises 1122, (a) fin...
 6.6.17: Linear Transformations In Exercises 922, determinewhether the funct...
 6.6.226: Similar Matrices In Exercises 1518, use the matrixto show that the ...
 6.6.265: Finding Fixed Points of a Linear Transformation InExercises 1522, f...
 6.6.333: Linear Transformation Given by a Matrix In Exercises1724, define th...
 6.6.102: Finding Bases for the Kernel and Range InExercises 1118, represents...
 6.6.169: Finding the Standard Matrix and the Image InExercises 1122, (a) fin...
 6.6.18: Linear Transformations In Exercises 922, determinewhether the funct...
 6.6.227: Similar Matrices In Exercises 1518, use the matrixto show that the ...
 6.6.266: Finding Fixed Points of a Linear Transformation InExercises 1522, f...
 6.6.334: Linear Transformation Given by a Matrix In Exercises1724, define th...
 6.6.103: Finding the Kernel, Nullity, Range, and Rank InExercises 1930, defi...
 6.6.170: Finding the Standard Matrix and the Image InExercises 1122, (a) fin...
 6.6.19: Linear Transformations In Exercises 922, determinewhether the funct...
 6.6.228: Diagonal Matrix for a Linear Transformation InExercises 19 and 20, ...
 6.6.267: Finding Fixed Points of a Linear Transformation InExercises 1522, f...
 6.6.335: Linear Transformation Given by a Matrix In Exercises1724, define th...
 6.6.104: Finding the Kernel, Nullity, Range, and Rank InExercises 1930, defi...
 6.6.171: Finding the Standard Matrix and the Image InExercises 1122, (a) fin...
 6.6.20: Linear Transformations In Exercises 922, determinewhether the funct...
 6.6.229: Diagonal Matrix for a Linear Transformation InExercises 19 and 20, ...
 6.6.268: Finding Fixed Points of a Linear Transformation InExercises 1522, f...
 6.6.336: Linear Transformation Given by a Matrix In Exercises1724, define th...
 6.6.105: Finding the Kernel, Nullity, Range, and Rank InExercises 1930, defi...
 6.6.172: Finding the Standard Matrix and the Image InExercises 1122, (a) fin...
 6.6.21: Linear Transformations In Exercises 922, determinewhether the funct...
 6.6.230: Proof Prove that if and are similar, then Is the converse true?
 6.6.269: Finding Fixed Points of a Linear Transformation InExercises 1522, f...
 6.6.337: Linear Transformation Given by a Matrix In Exercises1724, define th...
 6.6.106: Finding the Kernel, Nullity, Range, and Rank InExercises 1930, defi...
 6.6.173: Finding the Standard Matrix and the Image InExercises 1122, (a) fin...
 6.6.22: Linear Transformations In Exercises 922, determinewhether the funct...
 6.6.231: . Illustrate the result of Exercise 21 using the matrices where B P...
 6.6.270: Sketching an Image of the Unit Square In Exercises2328, sketch the ...
 6.6.338: Linear Transformation Given by a Matrix In Exercises1724, define th...
 6.6.107: Finding the Kernel, Nullity, Range, and Rank InExercises 1930, defi...
 6.6.174: Finding the Standard Matrix and the Image InExercises 2326, (a) fin...
 6.6.23: Let be a linear transformation from into suchthat and Findand T2, 1.T1
 6.6.232: Proof Prove that if and are similar, then thereexists a matrix such...
 6.6.271: Sketching an Image of the Unit Square In Exercises2328, sketch the ...
 6.6.339: Linear Transformation Given by a Matrix In Exercises1724, define th...
 6.6.108: Finding the Kernel, Nullity, Range, and Rank InExercises 1930, defi...
 6.6.175: Finding the Standard Matrix and the Image InExercises 2326, (a) fin...
 6.6.24: Let be a linear transformation from into suchthat and Findand T0, 2.T
 6.6.233: Use the result of Exercise 23 to find wherefor the matrices
 6.6.272: Sketching an Image of the Unit Square In Exercises2328, sketch the ...
 6.6.340: Use the standard matrix for counterclockwise rotationin to rotate t...
 6.6.109: Finding the Kernel, Nullity, Range, and Rank InExercises 1930, defi...
 6.6.176: Finding the Standard Matrix and the Image InExercises 2326, (a) fin...
 6.6.25: Linear Transformation and Bases In Exercises2528, let be a linear t...
 6.6.234: . Determine all matrices that are similar to In n n .
 6.6.273: Sketching an Image of the Unit Square In Exercises2328, sketch the ...
 6.6.341: Rotate the triangle in Exercise 25 counterclockwiseabout the point ...
 6.6.110: Finding the Kernel, Nullity, Range, and Rank InExercises 1930, defi...
 6.6.177: Finding the Standard Matrix and the Image InExercises 2326, (a) fin...
 6.6.26: Linear Transformation and Bases In Exercises2528, let be a linear t...
 6.6.235: Proof Prove that if is idempotent and is similar tothen is idempote...
 6.6.274: Sketching an Image of the Unit Square In Exercises2328, sketch the ...
 6.6.342: Finding Bases for the Kernel and Range In Exercises2730, find a bas...
 6.6.111: Finding the Kernel, Nullity, Range, and Rank InExercises 1930, defi...
 6.6.178: Finding Standard Matrices for Compositions InExercises 2730, find t...
 6.6.27: Linear Transformation and Bases In Exercises2528, let be a linear t...
 6.6.236: Proof Let be an matrix such thatProve that if is similar to then B ...
 6.6.275: Sketching an Image of the Unit Square In Exercises2328, sketch the ...
 6.6.343: Finding Bases for the Kernel and Range In Exercises2730, find a bas...
 6.6.112: Finding the Kernel, Nullity, Range, and Rank InExercises 1930, defi...
 6.6.179: Finding Standard Matrices for Compositions InExercises 2730, find t...
 6.6.28: Linear Transformation and Bases In Exercises2528, let be a linear t...
 6.6.237: Proof Let Prove that if then PBP 1x x.
 6.6.276: Sketching an Image of a Rectangle In Exercises2934, sketch the imag...
 6.6.344: Finding Bases for the Kernel and Range In Exercises2730, find a bas...
 6.6.113: Finding the Kernel, Nullity, Range, and Rank InExercises 1930, defi...
 6.6.180: Finding Standard Matrices for Compositions InExercises 2730, find t...
 6.6.29: Linear Transformation and Bases In Exercises 2932,let be a linear t...
 6.6.238: . Proof Complete the proof of Theorem 6.13 byproving that if is sim...
 6.6.277: Sketching an Image of a Rectangle In Exercises2934, sketch the imag...
 6.6.345: Finding Bases for the Kernel and Range In Exercises2730, find a bas...
 6.6.114: Finding the Kernel, Nullity, Range, and Rank InExercises 1930, defi...
 6.6.181: Finding Standard Matrices for Compositions InExercises 2730, find t...
 6.6.30: Linear Transformation and Bases In Exercises 2932,let be a linear t...
 6.6.239: . Writing Suppose and are similar. Explain whythey have the same rank.
 6.6.278: Sketching an Image of a Rectangle In Exercises2934, sketch the imag...
 6.6.346: Finding the Kernel, Nullity, Range, and Rank InExercises 3134, defi...
 6.6.115: Finding the Nullity and Describing the Kernel andRange In Exercises...
 6.6.182: Finding the Inverse of a Linear TransformationIn Exercises 3136, de...
 6.6.31: Linear Transformation and Bases In Exercises 2932,let be a linear t...
 6.6.240: . Proof Prove that if and are similar, then andare similar.
 6.6.279: Sketching an Image of a Rectangle In Exercises2934, sketch the imag...
 6.6.347: Finding the Kernel, Nullity, Range, and Rank InExercises 3134, defi...
 6.6.116: Finding the Nullity and Describing the Kernel andRange In Exercises...
 6.6.183: Finding the Inverse of a Linear TransformationIn Exercises 3136, de...
 6.6.32: Linear Transformation and Bases In Exercises 2932,let be a linear t...
 6.6.241: Proof Prove that if and are similar and isnonsingular, then is also...
 6.6.280: Sketching an Image of a Rectangle In Exercises2934, sketch the imag...
 6.6.348: Finding the Kernel, Nullity, Range, and Rank InExercises 3134, defi...
 6.6.117: Finding the Nullity and Describing the Kernel andRange In Exercises...
 6.6.184: Finding the Inverse of a Linear TransformationIn Exercises 3136, de...
 6.6.33: Linear Transformation Given by a Matrix In Exercises3338, define th...
 6.6.242: Proof Let , where is an invertiblematrix. Prove that the matrix is ...
 6.6.281: Sketching an Image of a Rectangle In Exercises2934, sketch the imag...
 6.6.349: Finding the Kernel, Nullity, Range, and Rank InExercises 3134, defi...
 6.6.118: Finding the Nullity and Describing the Kernel andRange In Exercises...
 6.6.185: Finding the Inverse of a Linear TransformationIn Exercises 3136, de...
 6.6.34: Linear Transformation Given by a Matrix In Exercises3338, define th...
 6.6.243: Proof Let , whereand is a diagonal matrix with main diagonal entries
 6.6.282: Sketching an Image of a Figure In Exercises3538, sketch each of the...
 6.6.350: Given T: R nullityT 2, rankT. 5R3A
 6.6.119: Finding the Nullity and Describing the Kernel andRange In Exercises...
 6.6.186: Finding the Inverse of a Linear TransformationIn Exercises 3136, de...
 6.6.35: Linear Transformation Given by a Matrix In Exercises3338, define th...
 6.6.244: 5. Writing Let be a basis for thevector space let be the standard b...
 6.6.283: Sketching an Image of a Figure In Exercises3538, sketch each of the...
 6.6.351: Given T: P nullityT 4, rankT. 5P3
 6.6.120: Finding the Nullity and Describing the Kernel andRange In Exercises...
 6.6.187: Finding the Inverse of a Linear TransformationIn Exercises 3136, de...
 6.6.36: Linear Transformation Given by a Matrix In Exercises3338, define th...
 6.6.245: (a) Given two bases and for a vector space andthe matrix for the li...
 6.6.284: Sketching an Image of a Figure In Exercises3538, sketch each of the...
 6.6.352: Given T: P rankT 3, nullityT. 4R5
 6.6.121: Finding the Nullity and Describing the Kernel andRange In Exercises...
 6.6.188: Finding the Image Two Ways In Exercises 3742, findby using (a) the ...
 6.6.37: Linear Transformation Given by a Matrix In Exercises3338, define th...
 6.6.246: True or False? In Exercises 37 and 38, determinewhether each statem...
 6.6.285: Sketching an Image of a Figure In Exercises3538, sketch each of the...
 6.6.353: Given T: M rankT 3, nullityT. 2,2
 6.6.122: Finding the Nullity and Describing the Kernel andRange In Exercises...
 6.6.189: Finding the Image Two Ways In Exercises 3742, findby using (a) the ...
 6.6.38: Linear Transformation Given by a Matrix In Exercises3338, define th...
 6.6.247: (a) The matrix for a linear transformation relative tothe basis is ...
 6.6.286: . The linear transformation defined by a diagonalmatrix with positi...
 6.6.354: Finding a Power of a Standard Matrix In Exercises3942, find the ind...
 6.6.123: Finding the Nullity of a Linear Transformation InExercises 3942, fi...
 6.6.190: Finding the Image Two Ways In Exercises 3742, findby using (a) the ...
 6.6.39: For the linear transformation from Exercise 33, find(a) (b) the pre...
 6.6.287: Repeat Exercise 39 for the linear transformationdefined by
 6.6.355: Finding a Power of a Standard Matrix In Exercises3942, find the ind...
 6.6.124: Finding the Nullity of a Linear Transformation InExercises 3942, fi...
 6.6.191: Finding the Image Two Ways In Exercises 3742, findby using (a) the ...
 6.6.40: Writing For the linear transformation from Exercise 34,find (a) and...
 6.6.288: Giving a Geometric Description In Exercises 4146,give a geometric d...
 6.6.356: Finding a Power of a Standard Matrix In Exercises3942, find the ind...
 6.6.125: Finding the Nullity of a Linear Transformation InExercises 3942, fi...
 6.6.192: Finding the Image Two Ways In Exercises 3742, findby using (a) the ...
 6.6.41: For the linear transformation from Exercise 35, find(a) and (b) the...
 6.6.289: Giving a Geometric Description In Exercises 4146,give a geometric d...
 6.6.357: Finding a Power of a Standard Matrix In Exercises3942, find the ind...
 6.6.126: Finding the Nullity of a Linear Transformation InExercises 3942, fi...
 6.6.193: Finding the Image Two Ways In Exercises 3742, findby using (a) the ...
 6.6.42: For the linear transformation from Exercise 36, find(a) and (b) the...
 6.6.290: Giving a Geometric Description In Exercises 4146,give a geometric d...
 6.6.358: Finding Standard Matrices for Compositions InExercises 43 and 44, f...
 6.6.127: Verifying That Is OnetoOne and Onto In Exercises4346, verify that...
 6.6.194: Let be given by Find the matrixfor relative to the bases and
 6.6.43: For the linear transformation from Exercise 37, find(a) and (b) the...
 6.6.291: Giving a Geometric Description In Exercises 4146,give a geometric d...
 6.6.359: Finding Standard Matrices for Compositions InExercises 43 and 44, f...
 6.6.128: Verifying That Is OnetoOne and Onto In Exercises4346, verify that...
 6.6.195: Let be given by Find the matrixfor relative to the bases and
 6.6.44: For the linear transformation from Exercise 38, find(a) (b) the pre...
 6.6.292: Giving a Geometric Description In Exercises 4146,give a geometric d...
 6.6.360: Finding the Inverse of a Linear TransformationIn Exercises 4548, de...
 6.6.129: Verifying That Is OnetoOne and Onto In Exercises4346, verify that...
 6.6.196: Calculus Let be a basis for asubspace of the space of continuous fu...
 6.6.45: . Let be a linear transformation from into suchthatFind (a) for (b)...
 6.6.293: Giving a Geometric Description In Exercises 4146,give a geometric d...
 6.6.361: Finding the Inverse of a Linear TransformationIn Exercises 4548, de...
 6.6.130: Verifying That Is OnetoOne and Onto In Exercises4346, verify that...
 6.6.197: Calculus Repeat Exercise 45 for B e2x, xe2x, x 2e2x.D
 6.6.46: . For the linear transformation from Exercise 45, letand find the p...
 6.6.294: Giving a Geometric Description In Exercises 47 and 48,give a geomet...
 6.6.362: Finding the Inverse of a Linear TransformationIn Exercises 4548, de...
 6.6.131: Determining Whether Is OnetoOne, Onto, orNeither In Exercises 475...
 6.6.198: Calculus Use the matrix from Exercise 45 to evaluate Dx3x 2xe x.
 6.6.47: . Find the inverse of the matrix given in Example 7. Whatlinear tra...
 6.6.295: Giving a Geometric Description In Exercises 47 and 48,give a geomet...
 6.6.363: Finding the Inverse of a Linear TransformationIn Exercises 4548, de...
 6.6.132: Determining Whether Is OnetoOne, Onto, orNeither In Exercises 475...
 6.6.199: Calculus Use the matrix from Exercise 46 to evaluate Dx3x 2xe x.
 6.6.48: . For the linear transformation given by find and such that T12, 5 ...
 6.6.296: Finding a Matrix to Produce a Rotation In Exercises4952, find the m...
 6.6.364: OnetoOne, Onto, and Invertible TransformationsIn Exercises 4952, ...
 6.6.133: Determining Whether Is OnetoOne, Onto, orNeither In Exercises 475...
 6.6.200: Calculus Let be a basis for andlet be the linear transformation rep...
 6.6.49: Projection in In Exercises 49 and 50, let the matrixrepresent the l...
 6.6.297: Finding a Matrix to Produce a Rotation In Exercises4952, find the m...
 6.6.365: OnetoOne, Onto, and Invertible TransformationsIn Exercises 4952, ...
 6.6.134: Determining Whether Is OnetoOne, Onto, orNeither In Exercises 475...
 6.6.201: Calculus Let be a basis for andlet be the linear transformation rep...
 6.6.50: Projection in In Exercises 49 and 50, let the matrixrepresent the l...
 6.6.298: Finding a Matrix to Produce a Rotation In Exercises4952, find the m...
 6.6.366: OnetoOne, Onto, and Invertible TransformationsIn Exercises 4952, ...
 6.6.135: Identify the zero element and standard basis for each ofthe isomorp...
 6.6.202: Define by(a) Find the matrix for relative to the standard basesfor ...
 6.6.51: Linear Transformation Given by a Matrix In Exercises5154, determine...
 6.6.299: Finding a Matrix to Produce a Rotation In Exercises4952, find the m...
 6.6.367: OnetoOne, Onto, and Invertible TransformationsIn Exercises 4952, ...
 6.6.136: Which vector spaces are isomorphic to R6?
 6.6.203: Let be a linear transformation such that forin Find the standard ma...
 6.6.52: Linear Transformation Given by a Matrix In Exercises5154, determine...
 6.6.300: Finding the Image of a Vector In Exercises 5356,find the image of t...
 6.6.368: Finding the Image Two Ways In Exercises 53 and 54,find by using (a)...
 6.6.137: Calculus Define by What is the What is the kernel of T?
 6.6.204: True or False? In Exercises 53 and 54, determinewhether each statem...
 6.6.53: Linear Transformation Given by a Matrix In Exercises5154, determine...
 6.6.301: Finding the Image of a Vector In Exercises 5356,find the image of t...
 6.6.369: Finding the Image Two Ways In Exercises 53 and 54,find by using (a)...
 6.6.138: Calculus Define by What is the kernel of T?
 6.6.205: (a) The composition of linear transformationsand represented by is ...
 6.6.54: Linear Transformation Given by a Matrix In Exercises5154, determine...
 6.6.302: Finding the Image of a Vector In Exercises 5356,find the image of t...
 6.6.370: Finding a Matrix for a Linear Transformation InExercises 55 and 56,...
 6.6.139: Let be the linear transformation that projects u v 2, 1, 1. (a) Fin...
 6.6.206: Guided Proof Let and beonetoone linear transformations. Prove tha...
 6.6.55: Let be a linear transformation from into suchthat andFind T2 6x x 2.Tx
 6.6.303: Finding the Image of a Vector In Exercises 5356,find the image of t...
 6.6.371: Finding a Matrix for a Linear Transformation InExercises 55 and 56,...
 6.6.140: Repeat Exercise 55 for v = 3, 0, 4.T
 6.6.207: Proof Prove Theorem 6.12.
 6.6.56: Let be a linear transformation from intosuch that
 6.6.304: Determining a Rotation InExercises 5762, determinewhich single coun...
 6.6.372: Similar Matrices In Exercises 57 and 58, use thematrix to show that...
 6.6.141: 7. For the transformation represented bywhat can be said about the ...
 6.6.208: Writing Is it always preferable to use the standardbasis for Discus...
 6.6.57: Calculus In Exercises 5760, let be the lineartransformation from in...
 6.6.305: Determining a Rotation InExercises 5762, determinewhich single coun...
 6.6.373: Similar Matrices In Exercises 57 and 58, use thematrix to show that...
 6.6.142: Consider the lineartransformation represented bywhere(a) Find the d...
 6.6.209: . Writing Look back at Theorem 4.19 and rephrase it interms of what...
 6.6.58: Calculus In Exercises 5760, let be the lineartransformation from in...
 6.6.306: Determining a Rotation InExercises 5762, determinewhich single coun...
 6.6.374: Define by where(a) Find the standard matrix for(b) Let be the linea...
 6.6.143: . Define by Show thatthe kernel of is the set of symmetric matrices.
 6.6.59: Calculus In Exercises 5760, let be the lineartransformation from in...
 6.6.307: Determining a Rotation InExercises 5762, determinewhich single coun...
 6.6.375: Define by where(a) Find the standard matrix for and show that(b) Sh...
 6.6.144: Determine a relationship among and such thatis isomorphic to
 6.6.60: Calculus In Exercises 5760, let be the lineartransformation from in...
 6.6.308: Determining a Rotation InExercises 5762, determinewhich single coun...
 6.6.376: Let and be linear transformations from intoShow that and are both l...
 6.6.145: True or False? In Exercises 61 and 62, determinewhether each statem...
 6.6.61: Calculus In Exercises 6164, for the lineartransformation from Examp...
 6.6.309: Determining a Rotation InExercises 5762, determinewhich single coun...
 6.6.377: Proof Let such that whereis a matrix. (Such a transformation is cal...
 6.6.146: . (a) The dimension of a linear transformation from avector space i...
 6.6.62: Calculus In Exercises 6164, for the lineartransformation from Examp...
 6.6.310: Determining a Matrix to Produce a Pair of RotationsIn Exercises 636...
 6.6.378: Sum of Two Linear Transformations In Exercises 63and 64, consider t...
 6.6.147: Guided Proof Let be an invertible matrix.Prove that the linear tran...
 6.6.63: Calculus In Exercises 6164, for the lineartransformation from Examp...
 6.6.311: Determining a Matrix to Produce a Pair of RotationsIn Exercises 636...
 6.6.379: Give an example for each
 6.6.148: Proof Let be a linear transformation. Provethat is onetoone if an...
 6.6.64: Calculus In Exercises 6164, for the lineartransformation from Examp...
 6.6.312: Determining a Matrix to Produce a Pair of RotationsIn Exercises 636...
 6.6.380: . Proof Let such that(a) Prove that is a linear transformation.(b) ...
 6.6.149: . Proof Prove Theorem 6.7.
 6.6.65: Calculus Let be a linear transformation from intosuch that
 6.6.66: Calculus Let be the linear transformation frominto given by the int...
 6.6.313: Determining a Matrix to Produce a Pair of RotationsIn Exercises 636...
 6.6.381: Proof Letandbe linear transformations.(a) Prove that is contained i...
 6.6.150: Proof Let be a linear transformation, andlet be a subspace of Prove...
 6.6.67: True or False? In Exercises 67 and 68, determinewhether each statem...
 6.6.314: Determining a Matrix to Produce a Pair of RotationsIn Exercises 636...
 6.6.382: . Let be an inner product space. For a fixed nonzerovector in let b...
 6.6.151: Writing Let be a linear transformation.Explain the differences betw...
 6.6.68: True or False? In Exercises 67 and 68, determinewhether each statem...
 6.6.315: 8. Describe the transformationdefined by each matrix. Assume and ar...
 6.6.383: 8. Calculus Let be a basis for asubspace of the space of continuous...
 6.6.69: . Writing Suppose such thatand(a) Determine for in(b) Give a geomet...
 6.6.384: Writing Are the vector spaces andexactly the same? Describe their s...
 6.6.70: 0. Writing Suppose such thatand(a) Determine for in(b) Give a geome...
 6.6.385: Calculus Define byFind the rank and nullity of T.
 6.6.71: 0. Writing Suppose such thatand(a) Determine for in(b) Give a geome...
 6.6.386: Identifying and Representing a Transformation InExercises 7176, (a)...
 6.6.72: Writing Find and from Exercise 71and give geometric descriptions of...
 6.6.387: Identifying and Representing a Transformation InExercises 7176, (a)...
 6.6.73: 3. Show that from Exercise 71 is represented by thematrix
 6.6.388: Identifying and Representing a Transformation InExercises 7176, (a)...
 6.6.74: Explain how to determine whether a function T: V W is a linear tra...
 6.6.389: Identifying and Representing a Transformation InExercises 7176, (a)...
 6.6.75: . Proof Use the concept of a fixed point of a lineartransformation ...
 6.6.390: Identifying and Representing a Transformation InExercises 7176, (a)...
 6.6.76: A translation in is a function of the formwhere at least one of the...
 6.6.391: Identifying and Representing a Transformation InExercises 7176, (a)...
 6.6.77: Proof Prove that (a) the zero transformation and (b)the identity tr...
 6.6.392: Sketching an Image of a Triangle In Exercises 7780,sketch the image...
 6.6.78: Let be a set of linearly independentvectors in Find a linear transf...
 6.6.393: Sketching an Image of a Triangle In Exercises 7780,sketch the image...
 6.6.79: Proof Let be a set of linearlydependent vectors in and let be a lin...
 6.6.394: Sketching an Image of a Triangle In Exercises 7780,sketch the image...
 6.6.80: Proof Let be an inner product space. For a fixedvector in define by...
 6.6.395: Sketching an Image of a Triangle In Exercises 7780,sketch the image...
 6.6.81: . Proof Define by(the trace of ). Prove that is a linear transforma...
 6.6.396: Giving a Geometric Description In Exercises 81 and 82,give a geomet...
 6.6.82: Let be an inner product space with a subspacehaving as an orthonorm...
 6.6.397: Giving a Geometric Description In Exercises 81 and 82,give a geomet...
 6.6.83: . Guided Proof Let be a basis for avector space Prove that if a lin...
 6.6.398: Finding a Matrix to Produce a Rotation In Exercises8386, find the m...
 6.6.84: Guided Proof Prove that is a lineartransformation if and only iffor...
 6.6.399: Finding a Matrix to Produce a Rotation In Exercises8386, find the m...
 6.6.400: Finding a Matrix to Produce a Rotation In Exercises8386, find the m...
 6.6.401: Finding a Matrix to Produce a Rotation In Exercises8386, find the m...
 6.6.402: Determining a Matrix to Produce a Pair of RotationsIn Exercises 879...
 6.6.403: Determining a Matrix to Produce a Pair of RotationsIn Exercises 879...
 6.6.404: Determining a Matrix to Produce a Pair of RotationsIn Exercises 879...
 6.6.405: Determining a Matrix to Produce a Pair of RotationsIn Exercises 879...
 6.6.406: Finding an Image of a Unit Cube In Exercises 9194,find the image of...
 6.6.407: Finding an Image of a Unit Cube In Exercises 9194,find the image of...
 6.6.408: Finding an Image of a Unit Cube In Exercises 9194,find the image of...
 6.6.409: Finding an Image of a Unit Cube In Exercises 9194,find the image of...
 6.6.410: True or False? In Exercises 9598, determine whethereach statement i...
 6.6.411: (a) Linear transformations called reflections that map apoint in th...
 6.6.412: . (a) In calculus, any linear function is also a lineartransformati...
 6.6.413: (a) For polynomials, the differential operator is alinear transform...
Solutions for Chapter 6: Inner Product Spaces
Full solutions for Elementary Linear Algebra  7th Edition
ISBN: 9781133110873
Solutions for Chapter 6: Inner Product Spaces
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 424 problems in chapter 6: Inner Product Spaces have been answered, more than 11304 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra was written by and is associated to the ISBN: 9781133110873. Chapter 6: Inner Product Spaces includes 424 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 7.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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 .

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.