 4.4.1: Linear Combinations In Exercises 14, write each vector as a linear ...
 4.4.4.4.1: Linear Combinations In Exercises 14, write each vector as a linear ...
 4.4.2: Linear Combinations In Exercises 14, write each vector as a linear ...
 4.4.4.4.2: Linear Combinations In Exercises 14, write each vector as a linear ...
 4.4.3: Linear Combinations In Exercises 14, write each vector as a linear ...
 4.4.4.4.3: Linear Combinations In Exercises 14, write each vector as a linear ...
 4.4.4: Linear Combinations In Exercises 14, write each vector as a linear ...
 4.4.4.4.4: Linear Combinations In Exercises 14, write each vector as a linear ...
 4.4.5: Linear Combinations In Exercises 58, for the matricesA = [2431] and...
 4.4.4.4.5: Linear Combinations In Exercises 58, for the matricesA = [2431] and...
 4.4.6: Linear Combinations In Exercises 58, for the matricesA = [2431] and...
 4.4.4.4.6: Linear Combinations In Exercises 58, for the matricesA = [2431] and...
 4.4.7: Linear Combinations In Exercises 58, for the matricesA = [2431] and...
 4.4.4.4.7: Linear Combinations In Exercises 58, for the matricesA = [2431] and...
 4.4.8: Linear Combinations In Exercises 58, for the matricesA = [2431] and...
 4.4.4.4.8: Linear Combinations In Exercises 58, for the matricesA = [2431] and...
 4.4.9: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.4.4.9: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.10: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.4.4.10: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.11: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.4.4.11: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.12: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.4.4.12: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.13: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.4.4.13: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.14: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.4.4.14: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.15: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.4.4.15: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.16: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.4.4.16: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.17: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.4.4.17: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.18: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.4.4.18: Spanning Sets In Exercises 918, determine whether the set S spans R...
 4.4.19: Spanning Sets In Exercises 1924, determine whether the set S spans ...
 4.4.4.4.19: Spanning Sets In Exercises 1924, determine whether the set S spans ...
 4.4.20: Spanning Sets In Exercises 1924, determine whether the set S spans ...
 4.4.4.4.20: Spanning Sets In Exercises 1924, determine whether the set S spans ...
 4.4.21: Spanning Sets In Exercises 1924, determine whether the set S spans ...
 4.4.4.4.21: Spanning Sets In Exercises 1924, determine whether the set S spans ...
 4.4.22: Spanning Sets In Exercises 1924, determine whether the set S spans ...
 4.4.4.4.22: Spanning Sets In Exercises 1924, determine whether the set S spans ...
 4.4.23: Spanning Sets In Exercises 1924, determine whether the set S spans ...
 4.4.4.4.23: Spanning Sets In Exercises 1924, determine whether the set S spans ...
 4.4.24: Spanning Sets In Exercises 1924, determine whether the set S spans ...
 4.4.4.4.24: Spanning Sets In Exercises 1924, determine whether the set S spans ...
 4.4.25: Determine whether the set S = {1, x2, 2 + x2} spans P2.
 4.4.4.4.25: Determine whether the set S = {1, x2, 2 + x2} spans P2.
 4.4.26: Determine whether the setS = {2x + x2, 8 + x3, x2 + x3, 4 + x2}span...
 4.4.4.4.26: Determine whether the setS = {2x + x2, 8 + x3, x2 + x3, 4 + x2}span...
 4.4.27: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.4.4.27: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.28: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.4.4.28: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.29: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.4.4.29: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.30: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.4.4.30: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.31: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.4.4.31: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.32: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.4.4.32: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.33: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.4.4.33: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.34: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.4.4.34: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.35: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.4.4.35: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.36: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.4.4.36: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.37: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.4.4.37: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.38: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.4.4.38: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.39: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.4.4.39: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.40: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.4.4.40: Testing for Linear Independence In Exercises 2740, determine whethe...
 4.4.41: Testing for Linear Independence In Exercises 4148, determine whethe...
 4.4.4.4.41: Testing for Linear Independence In Exercises 4148, determine whethe...
 4.4.42: Testing for Linear Independence In Exercises 4148, determine whethe...
 4.4.4.4.42: Testing for Linear Independence In Exercises 4148, determine whethe...
 4.4.43: Testing for Linear Independence In Exercises 4148, determine whethe...
 4.4.4.4.43: Testing for Linear Independence In Exercises 4148, determine whethe...
 4.4.44: Testing for Linear Independence In Exercises 4148, determine whethe...
 4.4.4.4.44: Testing for Linear Independence In Exercises 4148, determine whethe...
 4.4.45: Testing for Linear Independence In Exercises 4148, determine whethe...
 4.4.4.4.45: Testing for Linear Independence In Exercises 4148, determine whethe...
 4.4.46: Testing for Linear Independence In Exercises 4148, determine whethe...
 4.4.4.4.46: Testing for Linear Independence In Exercises 4148, determine whethe...
 4.4.47: Testing for Linear Independence In Exercises 4148, determine whethe...
 4.4.4.4.47: Testing for Linear Independence In Exercises 4148, determine whethe...
 4.4.48: Testing for Linear Independence In Exercises 4148, determine whethe...
 4.4.4.4.48: Testing for Linear Independence In Exercises 4148, determine whethe...
 4.4.49: Testing for Linear Independence In Exercises 4952, determine whethe...
 4.4.4.4.49: Testing for Linear Independence In Exercises 4952, determine whethe...
 4.4.50: Testing for Linear Independence In Exercises 4952, determine whethe...
 4.4.4.4.50: Testing for Linear Independence In Exercises 4952, determine whethe...
 4.4.51: Testing for Linear Independence In Exercises 4952, determine whethe...
 4.4.4.4.51: Testing for Linear Independence In Exercises 4952, determine whethe...
 4.4.52: Testing for Linear Independence In Exercises 4952, determine whethe...
 4.4.4.4.52: Testing for Linear Independence In Exercises 4952, determine whethe...
 4.4.53: Showing Linear Dependence In Exercises 5356, show that the set is l...
 4.4.4.4.53: Showing Linear Dependence In Exercises 5356, show that the set is l...
 4.4.54: Showing Linear Dependence In Exercises 5356, show that the set is l...
 4.4.4.4.54: Showing Linear Dependence In Exercises 5356, show that the set is l...
 4.4.55: Showing Linear Dependence In Exercises 5356, show that the set is l...
 4.4.4.4.55: Showing Linear Dependence In Exercises 5356, show that the set is l...
 4.4.56: Showing Linear Dependence In Exercises 5356, show that the set is l...
 4.4.4.4.56: Showing Linear Dependence In Exercises 5356, show that the set is l...
 4.4.57: For which values of t is each set linearly independent?(a) S = {(t,...
 4.4.4.4.57: For which values of t is each set linearly independent?(a) S = {(t,...
 4.4.58: For which values of t is each set linearly independent?(a) S = {(t,...
 4.4.4.4.58: For which values of t is each set linearly independent?(a) S = {(t,...
 4.4.59: Proof Complete the proof of Theorem 4.7.
 4.4.4.4.59: Proof Complete the proof of Theorem 4.7.
 4.4.60: CAPSTONE By inspection, determine whyeach of the sets is linearly d...
 4.4.4.4.60: CAPSTONE By inspection, determine whyeach of the sets is linearly d...
 4.4.61: Spanning the Same Subspace In Exercises 61 and 62, show that the se...
 4.4.4.4.61: Spanning the Same Subspace In Exercises 61 and 62, show that the se...
 4.4.62: Spanning the Same Subspace In Exercises 61 and 62, show that the se...
 4.4.4.4.62: Spanning the Same Subspace In Exercises 61 and 62, show that the se...
 4.4.63: True or False? In Exercises 63 and 64, determine whether each state...
 4.4.4.4.63: True or False? In Exercises 63 and 64, determine whether each state...
 4.4.64: True or False? In Exercises 63 and 64, determine whether each state...
 4.4.4.4.64: True or False? In Exercises 63 and 64, determine whether each state...
 4.4.65: Proof In Exercises 65 and 66, prove that the set of vectors is line...
 4.4.4.4.65: Proof In Exercises 65 and 66, prove that the set of vectors is line...
 4.4.66: Proof In Exercises 65 and 66, prove that the set of vectors is line...
 4.4.4.4.66: Proof In Exercises 65 and 66, prove that the set of vectors is line...
 4.4.67: Guided Proof Prove that a nonempty subset of afinite set of linearl...
 4.4.4.4.67: Guided Proof Prove that a nonempty subset of afinite set of linearl...
 4.4.68: Proof Prove that if S1 is a nonempty subset of thefinite set S2, an...
 4.4.4.4.68: Proof Prove that if S1 is a nonempty subset of thefinite set S2, an...
 4.4.69: Proof Prove that any set of vectors containing the zero vector is l...
 4.4.4.4.69: Proof Prove that any set of vectors containing the zero vector is l...
 4.4.70: Proof When the set of vectors {u1, u2, . . . , un} islinearly indep...
 4.4.4.4.70: Proof When the set of vectors {u1, u2, . . . , un} islinearly indep...
 4.4.71: Proof Let {v1, v2, . . . , vk} be a linearly independentset of vect...
 4.4.4.4.71: Proof Let {v1, v2, . . . , vk} be a linearly independentset of vect...
 4.4.72: Proof When V is spanned by {v1, v2, . . . , vk}and one of these vec...
 4.4.4.4.72: Proof When V is spanned by {v1, v2, . . . , vk}and one of these vec...
 4.4.73: Proof Let S = {u, v} be a linearly independent set.Prove that the s...
 4.4.4.4.73: Proof Let S = {u, v} be a linearly independent set.Prove that the s...
 4.4.74: Let u, v, and w be any three vectors from a vectorspace V. Determin...
 4.4.4.4.74: Let u, v, and w be any three vectors from a vectorspace V. Determin...
 4.4.75: Proof Let A be a nonsingular matrix of order 3. Provethat if {v1, v...
 4.4.4.4.75: Proof Let A be a nonsingular matrix of order 3. Provethat if {v1, v...
 4.4.76: Let f1(x) = 3x and f2(x) = x. Graph both functionson the interval 2...
 4.4.4.4.76: Let f1(x) = 3x and f2(x) = x. Graph both functionson the interval 2...
 4.4.77: Proof Prove the corollary to Theorem 4.8: Twovectors u and v are li...
 4.4.4.4.77: Proof Prove the corollary to Theorem 4.8: Twovectors u and v are li...
Solutions for Chapter 4.4: Spanning Sets and Linear Independence
Full solutions for Elementary Linear Algebra  8th Edition
ISBN: 9781305658004
Solutions for Chapter 4.4: Spanning Sets and Linear Independence
Get Full SolutionsElementary Linear Algebra was written by and is associated to the ISBN: 9781305658004. Chapter 4.4: Spanning Sets and Linear Independence includes 154 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 154 problems in chapter 4.4: Spanning Sets and Linear Independence have been answered, more than 47701 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 8.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

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.

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.

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

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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 .

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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·

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.