 4.4.1: Vector Operations In Exercises 14, find (a) u + v, (b) 2v, (c) u v,...
 4.1: Vector Operations In Exercises 14, find (a) u + v, (b) 2v, (c) u v,...
 4.4.2: Vector Operations In Exercises 14, find (a) u + v, (b) 2v, (c) u v,...
 4.2: Vector Operations In Exercises 14, find (a) u + v, (b) 2v, (c) u v,...
 4.4.3: Vector Operations In Exercises 14, find (a) u + v, (b) 2v, (c) u v,...
 4.3: Vector Operations In Exercises 14, find (a) u + v, (b) 2v, (c) u v,...
 4.4.4: Vector Operations In Exercises 14, find (a) u + v, (b) 2v, (c) u v,...
 4.4: Vector Operations In Exercises 14, find (a) u + v, (b) 2v, (c) u v,...
 4.4.5: Solving a Vector Equation In Exercises 58, solve for x, where u = (...
 4.5: Solving a Vector Equation In Exercises 58, solve for x, where u = (...
 4.4.6: Solving a Vector Equation In Exercises 58, solve for x, where u = (...
 4.6: Solving a Vector Equation In Exercises 58, solve for x, where u = (...
 4.4.7: Solving a Vector Equation In Exercises 58, solve for x, where u = (...
 4.7: Solving a Vector Equation In Exercises 58, solve for x, where u = (...
 4.4.8: Solving a Vector Equation In Exercises 58, solve for x, where u = (...
 4.8: Solving a Vector Equation In Exercises 58, solve for x, where u = (...
 4.4.9: Writing a Linear Combination In Exercises 912, write v as a linear ...
 4.9: Writing a Linear Combination In Exercises 912, write v as a linear ...
 4.4.10: Writing a Linear Combination In Exercises 912, write v as a linear ...
 4.10: Writing a Linear Combination In Exercises 912, write v as a linear ...
 4.4.11: Writing a Linear Combination In Exercises 912, write v as a linear ...
 4.11: Writing a Linear Combination In Exercises 912, write v as a linear ...
 4.4.12: Writing a Linear Combination In Exercises 912, write v as a linear ...
 4.12: Writing a Linear Combination In Exercises 912, write v as a linear ...
 4.4.13: Describing the Zero Vector and the Additive Inverse In Exercises 13...
 4.13: Describing the Zero Vector and the Additive Inverse In Exercises 13...
 4.4.14: Describing the Zero Vector and the Additive Inverse In Exercises 13...
 4.14: Describing the Zero Vector and the Additive Inverse In Exercises 13...
 4.4.15: Describing the Zero Vector and the Additive Inverse In Exercises 13...
 4.15: Describing the Zero Vector and the Additive Inverse In Exercises 13...
 4.4.16: Describing the Zero Vector and the Additive Inverse In Exercises 13...
 4.16: Describing the Zero Vector and the Additive Inverse In Exercises 13...
 4.4.17: Determining Subspaces In Exercises 1724, determine whether W is a s...
 4.17: Determining Subspaces In Exercises 1724, determine whether W is a s...
 4.4.18: Determining Subspaces In Exercises 1724, determine whether W is a s...
 4.18: Determining Subspaces In Exercises 1724, determine whether W is a s...
 4.4.19: Determining Subspaces In Exercises 1724, determine whether W is a s...
 4.19: Determining Subspaces In Exercises 1724, determine whether W is a s...
 4.4.20: Determining Subspaces In Exercises 1724, determine whether W is a s...
 4.20: Determining Subspaces In Exercises 1724, determine whether W is a s...
 4.4.21: Determining Subspaces In Exercises 1724, determine whether W is a s...
 4.21: Determining Subspaces In Exercises 1724, determine whether W is a s...
 4.4.22: Determining Subspaces In Exercises 1724, determine whether W is a s...
 4.22: Determining Subspaces In Exercises 1724, determine whether W is a s...
 4.4.23: Determining Subspaces In Exercises 1724, determine whether W is a s...
 4.23: Determining Subspaces In Exercises 1724, determine whether W is a s...
 4.4.24: Determining Subspaces In Exercises 1724, determine whether W is a s...
 4.24: Determining Subspaces In Exercises 1724, determine whether W is a s...
 4.4.25: Which of the subsets of R3 is a subspace of R3?(a) W = {(x1, x2, x3...
 4.25: Which of the subsets of R3 is a subspace of R3?(a) W = {(x1, x2, x3...
 4.4.26: Which of the subsets of R3 is a subspace of R3?(a) W = {(x1, x2, x3...
 4.26: Which of the subsets of R3 is a subspace of R3?(a) W = {(x1, x2, x3...
 4.4.27: Spanning Sets, Linear Independence, and Bases In Exercises 2732, de...
 4.27: Spanning Sets, Linear Independence, and Bases In Exercises 2732, de...
 4.4.28: Spanning Sets, Linear Independence, and Bases In Exercises 2732, de...
 4.28: Spanning Sets, Linear Independence, and Bases In Exercises 2732, de...
 4.4.29: Spanning Sets, Linear Independence, and Bases In Exercises 2732, de...
 4.29: Spanning Sets, Linear Independence, and Bases In Exercises 2732, de...
 4.4.30: Spanning Sets, Linear Independence, and Bases In Exercises 2732, de...
 4.30: Spanning Sets, Linear Independence, and Bases In Exercises 2732, de...
 4.4.31: Spanning Sets, Linear Independence, and Bases In Exercises 2732, de...
 4.31: Spanning Sets, Linear Independence, and Bases In Exercises 2732, de...
 4.4.32: Spanning Sets, Linear Independence, and Bases In Exercises 2732, de...
 4.32: Spanning Sets, Linear Independence, and Bases In Exercises 2732, de...
 4.4.33: Determine whetherS = {1 t, 2t + 3t2, t2 2t3, 2 + t3}is a basis for P3
 4.33: Determine whetherS = {1 t, 2t + 3t2, t2 2t3, 2 + t3}is a basis for P3
 4.4.34: Determine whether S = {1, t, 1 + t 2} is a basis for P2.
 4.34: Determine whether S = {1, t, 1 + t 2} is a basis for P2.
 4.4.35: Determining Whether a Set Is a Basis In Exercises 35 and 36, determ...
 4.35: Determining Whether a Set Is a Basis In Exercises 35 and 36, determ...
 4.4.36: Determining Whether a Set Is a Basis In Exercises 35 and 36, determ...
 4.36: Determining Whether a Set Is a Basis In Exercises 35 and 36, determ...
 4.4.37: Finding the Nullspace, Nullity, and Rank of a Matrix In Exercises 3...
 4.37: Finding the Nullspace, Nullity, and Rank of a Matrix In Exercises 3...
 4.4.38: Finding the Nullspace, Nullity, and Rank of a Matrix In Exercises 3...
 4.38: Finding the Nullspace, Nullity, and Rank of a Matrix In Exercises 3...
 4.4.39: Finding the Nullspace, Nullity, and Rank of a Matrix In Exercises 3...
 4.39: Finding the Nullspace, Nullity, and Rank of a Matrix In Exercises 3...
 4.4.40: Finding the Nullspace, Nullity, and Rank of a Matrix In Exercises 3...
 4.40: Finding the Nullspace, Nullity, and Rank of a Matrix In Exercises 3...
 4.4.41: Finding the Nullspace, Nullity, and Rank of a Matrix In Exercises 3...
 4.41: Finding the Nullspace, Nullity, and Rank of a Matrix In Exercises 3...
 4.4.42: Finding the Nullspace, Nullity, and Rank of a Matrix In Exercises 3...
 4.42: Finding the Nullspace, Nullity, and Rank of a Matrix In Exercises 3...
 4.4.43: Finding a Basis for a Row Space and Rank In Exercises 4346, find (a...
 4.43: Finding a Basis for a Row Space and Rank In Exercises 4346, find (a...
 4.4.44: Finding a Basis for a Row Space and Rank In Exercises 4346, find (a...
 4.44: Finding a Basis for a Row Space and Rank In Exercises 4346, find (a...
 4.4.45: Finding a Basis for a Row Space and Rank In Exercises 4346, find (a...
 4.45: Finding a Basis for a Row Space and Rank In Exercises 4346, find (a...
 4.4.46: Finding a Basis for a Row Space and Rank In Exercises 4346, find (a...
 4.46: Finding a Basis for a Row Space and Rank In Exercises 4346, find (a...
 4.4.47: Finding a Basis and Dimension In Exercises 4750, find (a) a basis f...
 4.47: Finding a Basis and Dimension In Exercises 4750, find (a) a basis f...
 4.4.48: Finding a Basis and Dimension In Exercises 4750, find (a) a basis f...
 4.48: Finding a Basis and Dimension In Exercises 4750, find (a) a basis f...
 4.4.49: Finding a Basis and Dimension In Exercises 4750, find (a) a basis f...
 4.49: Finding a Basis and Dimension In Exercises 4750, find (a) a basis f...
 4.4.50: Finding a Basis and Dimension In Exercises 4750, find (a) a basis f...
 4.50: Finding a Basis and Dimension In Exercises 4750, find (a) a basis f...
 4.4.51: Finding a Coordinate Matrix In Exercises 5156, given the coordinate...
 4.51: Finding a Coordinate Matrix In Exercises 5156, given the coordinate...
 4.4.52: Finding a Coordinate Matrix In Exercises 5156, given the coordinate...
 4.52: Finding a Coordinate Matrix In Exercises 5156, given the coordinate...
 4.4.53: Finding a Coordinate Matrix In Exercises 5156, given the coordinate...
 4.53: Finding a Coordinate Matrix In Exercises 5156, given the coordinate...
 4.4.54: Finding a Coordinate Matrix In Exercises 5156, given the coordinate...
 4.54: Finding a Coordinate Matrix In Exercises 5156, given the coordinate...
 4.4.55: Finding a Coordinate Matrix In Exercises 5156, given the coordinate...
 4.55: Finding a Coordinate Matrix In Exercises 5156, given the coordinate...
 4.4.56: Finding a Coordinate Matrix In Exercises 5156, given the coordinate...
 4.56: Finding a Coordinate Matrix In Exercises 5156, given the coordinate...
 4.4.57: Finding a Coordinate Matrix In Exercises 5762, find the coordinate ...
 4.57: Finding a Coordinate Matrix In Exercises 5762, find the coordinate ...
 4.4.58: Finding a Coordinate Matrix In Exercises 5762, find the coordinate ...
 4.58: Finding a Coordinate Matrix In Exercises 5762, find the coordinate ...
 4.4.59: Finding a Coordinate Matrix In Exercises 5762, find the coordinate ...
 4.59: Finding a Coordinate Matrix In Exercises 5762, find the coordinate ...
 4.4.60: Finding a Coordinate Matrix In Exercises 5762, find the coordinate ...
 4.60: Finding a Coordinate Matrix In Exercises 5762, find the coordinate ...
 4.4.61: Finding a Coordinate Matrix In Exercises 5762, find the coordinate ...
 4.61: Finding a Coordinate Matrix In Exercises 5762, find the coordinate ...
 4.4.62: Finding a Coordinate Matrix In Exercises 5762, find the coordinate ...
 4.62: Finding a Coordinate Matrix In Exercises 5762, find the coordinate ...
 4.4.63: Finding a Transition Matrix In Exercises 6368, find the transition ...
 4.63: Finding a Transition Matrix In Exercises 6368, find the transition ...
 4.4.64: Finding a Transition Matrix In Exercises 6368, find the transition ...
 4.64: Finding a Transition Matrix In Exercises 6368, find the transition ...
 4.4.65: Finding a Transition Matrix In Exercises 6368, find the transition ...
 4.65: Finding a Transition Matrix In Exercises 6368, find the transition ...
 4.4.66: Finding a Transition Matrix In Exercises 6368, find the transition ...
 4.66: Finding a Transition Matrix In Exercises 6368, find the transition ...
 4.4.67: Finding a Transition Matrix In Exercises 6368, find the transition ...
 4.67: Finding a Transition Matrix In Exercises 6368, find the transition ...
 4.4.68: Finding a Transition Matrix In Exercises 6368, find the transition ...
 4.68: Finding a Transition Matrix In Exercises 6368, find the transition ...
 4.4.69: Finding Transition and Coordinate Matrices In Exercises 6972, (a) f...
 4.69: Finding Transition and Coordinate Matrices In Exercises 6972, (a) f...
 4.4.70: Finding Transition and Coordinate Matrices In Exercises 6972, (a) f...
 4.70: Finding Transition and Coordinate Matrices In Exercises 6972, (a) f...
 4.4.71: Finding Transition and Coordinate Matrices In Exercises 6972, (a) f...
 4.71: Finding Transition and Coordinate Matrices In Exercises 6972, (a) f...
 4.4.72: Finding Transition and Coordinate Matrices In Exercises 6972, (a) f...
 4.72: Finding Transition and Coordinate Matrices In Exercises 6972, (a) f...
 4.4.73: Let W be the subspace of P3 [the set of all polynomials p(x) of deg...
 4.73: Let W be the subspace of P3 [the set of all polynomials p(x) of deg...
 4.4.74: Calculus Let V = C(, ), the vector space ofall continuously differe...
 4.74: Calculus Let V = C(, ), the vector space ofall continuously differe...
 4.4.75: Writing Let B = {p1(x), p2(x), . . . , pn(x), pn+1(x)}be a basis fo...
 4.75: Writing Let B = {p1(x), p2(x), . . . , pn(x), pn+1(x)}be a basis fo...
 4.4.76: Proof Let A and B be n n square matrices with A O and B O. Prove th...
 4.76: Proof Let A and B be n n square matrices with A O and B O. Prove th...
 4.4.77: Proof Let V = P5 and consider the set W of allpolynomials of the fo...
 4.77: Proof Let V = P5 and consider the set W of allpolynomials of the fo...
 4.4.78: Let v1, v2, and v3 be three linearly independentvectors in a vector...
 4.78: Let v1, v2, and v3 be three linearly independentvectors in a vector...
 4.4.79: Proof Let A be an n n square matrix. Prove that the row vectors of ...
 4.79: Proof Let A be an n n square matrix. Prove that the row vectors of ...
 4.4.80: Proof Let A be an n n square matrix, and let be ascalar. Prove that...
 4.80: Proof Let A be an n n square matrix, and let be ascalar. Prove that...
 4.4.81: Let f(x) = x and g(x) = x.(a) Show that f and g are linearly indepe...
 4.81: Let f(x) = x and g(x) = x.(a) Show that f and g are linearly indepe...
 4.4.82: Describe how the domain of a set of functions can influence whether...
 4.82: Describe how the domain of a set of functions can influence whether...
 4.4.83: True or False? In Exercises 8386, determine whether each statement ...
 4.83: True or False? In Exercises 8386, determine whether each statement ...
 4.4.84: True or False? In Exercises 8386, determine whether each statement ...
 4.84: True or False? In Exercises 8386, determine whether each statement ...
 4.4.85: True or False? In Exercises 8386, determine whether each statement ...
 4.85: True or False? In Exercises 8386, determine whether each statement ...
 4.4.86: True or False? In Exercises 8386, determine whether each statement ...
 4.86: True or False? In Exercises 8386, determine whether each statement ...
 4.4.87: Determining Solutions of a Differential Equation In Exercises 8790,...
 4.87: Determining Solutions of a Differential Equation In Exercises 8790,...
 4.4.88: Determining Solutions of a Differential Equation In Exercises 8790,...
 4.88: Determining Solutions of a Differential Equation In Exercises 8790,...
 4.4.89: Determining Solutions of a Differential Equation In Exercises 8790,...
 4.89: Determining Solutions of a Differential Equation In Exercises 8790,...
 4.4.90: Determining Solutions of a Differential Equation In Exercises 8790,...
 4.90: Determining Solutions of a Differential Equation In Exercises 8790,...
 4.4.91: Finding the Wronskian for a Set of Functions In Exercises 9194, fin...
 4.91: Finding the Wronskian for a Set of Functions In Exercises 9194, fin...
 4.4.92: Finding the Wronskian for a Set of Functions In Exercises 9194, fin...
 4.92: Finding the Wronskian for a Set of Functions In Exercises 9194, fin...
 4.4.93: Finding the Wronskian for a Set of Functions In Exercises 9194, fin...
 4.93: Finding the Wronskian for a Set of Functions In Exercises 9194, fin...
 4.4.94: Finding the Wronskian for a Set of Functions In Exercises 9194, fin...
 4.94: Finding the Wronskian for a Set of Functions In Exercises 9194, fin...
 4.4.95: Testing for Linear Independence In Exercises 9598, (a) verify that ...
 4.95: Testing for Linear Independence In Exercises 9598, (a) verify that ...
 4.4.96: Testing for Linear Independence In Exercises 9598, (a) verify that ...
 4.96: Testing for Linear Independence In Exercises 9598, (a) verify that ...
 4.4.97: Testing for Linear Independence In Exercises 9598, (a) verify that ...
 4.97: Testing for Linear Independence In Exercises 9598, (a) verify that ...
 4.4.98: Testing for Linear Independence In Exercises 9598, (a) verify that ...
 4.98: Testing for Linear Independence In Exercises 9598, (a) verify that ...
 4.4.99: Identifying and Graphing a Conic Section In Exercises 99106, identi...
 4.99: Identifying and Graphing a Conic Section In Exercises 99106, identi...
 4.4.100: Identifying and Graphing a Conic Section In Exercises 99106, identi...
 4.100: Identifying and Graphing a Conic Section In Exercises 99106, identi...
 4.4.101: Identifying and Graphing a Conic Section In Exercises 99106, identi...
 4.101: Identifying and Graphing a Conic Section In Exercises 99106, identi...
 4.4.102: Identifying and Graphing a Conic Section In Exercises 99106, identi...
 4.102: Identifying and Graphing a Conic Section In Exercises 99106, identi...
 4.4.103: Identifying and Graphing a Conic Section In Exercises 99106, identi...
 4.103: Identifying and Graphing a Conic Section In Exercises 99106, identi...
 4.4.104: Identifying and Graphing a Conic Section In Exercises 99106, identi...
 4.104: Identifying and Graphing a Conic Section In Exercises 99106, identi...
 4.4.105: Identifying and Graphing a Conic Section In Exercises 99106, identi...
 4.105: Identifying and Graphing a Conic Section In Exercises 99106, identi...
 4.4.106: Identifying and Graphing a Conic Section In Exercises 99106, identi...
 4.106: Identifying and Graphing a Conic Section In Exercises 99106, identi...
 4.4.107: Rotation of a Conic Section In Exercises 107110, perform a rotation...
 4.107: Rotation of a Conic Section In Exercises 107110, perform a rotation...
 4.108: Rotation of a Conic Section In Exercises 107110, perform a rotation...
 4.4.108: Rotation of a Conic Section In Exercises 107110, perform a rotation...
 4.109: Rotation of a Conic Section In Exercises 107110, perform a rotation...
 4.4.109: Rotation of a Conic Section In Exercises 107110, perform a rotation...
 4.110: Rotation of a Conic Section In Exercises 107110, perform a rotation...
 4.4.110: Rotation of a Conic Section In Exercises 107110, perform a rotation...
Solutions for Chapter 4: Vector Spaces
Full solutions for Elementary Linear Algebra  8th Edition
ISBN: 9781305658004
Solutions for Chapter 4: Vector Spaces
Get Full SolutionsSince 220 problems in chapter 4: Vector Spaces have been answered, more than 47842 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Linear Algebra was written by and is associated to the ISBN: 9781305658004. Chapter 4: Vector Spaces includes 220 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 8.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).