 4.3.4.3.1: Verifying Subspaces In Exercises 16, verify that W is a subspace of...
 4.3.1: Verifying Subspaces In Exercises 16, verify that W is a subspace of...
 4.3.4.3.2: Verifying Subspaces In Exercises 16, verify that W is a subspace of...
 4.3.2: Verifying Subspaces In Exercises 16, verify that W is a subspace of...
 4.3.4.3.3: Verifying Subspaces In Exercises 16, verify that W is a subspace of...
 4.3.3: Verifying Subspaces In Exercises 16, verify that W is a subspace of...
 4.3.4.3.4: Verifying Subspaces In Exercises 16, verify that W is a subspace of...
 4.3.4: Verifying Subspaces In Exercises 16, verify that W is a subspace of...
 4.3.4.3.5: Verifying Subspaces In Exercises 16, verify that W is a subspace of...
 4.3.5: Verifying Subspaces In Exercises 16, verify that W is a subspace of...
 4.3.4.3.6: Verifying Subspaces In Exercises 16, verify that W is a subspace of...
 4.3.6: Verifying Subspaces In Exercises 16, verify that W is a subspace of...
 4.3.4.3.7: Subsets That Are Not Subspaces In Exercises 720, W is not a subspac...
 4.3.7: Subsets That Are Not Subspaces In Exercises 720, W is not a subspac...
 4.3.4.3.8: Subsets That Are Not Subspaces In Exercises 720, W is not a subspac...
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 4.3.4.3.9: Subsets That Are Not Subspaces In Exercises 720, W is not a subspac...
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 4.3.10: Subsets That Are Not Subspaces In Exercises 720, W is not a subspac...
 4.3.4.3.11: Subsets That Are Not Subspaces In Exercises 720, W is not a subspac...
 4.3.11: Subsets That Are Not Subspaces In Exercises 720, W is not a subspac...
 4.3.4.3.12: Subsets That Are Not Subspaces In Exercises 720, W is not a subspac...
 4.3.12: Subsets That Are Not Subspaces In Exercises 720, W is not a subspac...
 4.3.4.3.13: Subsets That Are Not Subspaces In Exercises 720, W is not a subspac...
 4.3.13: Subsets That Are Not Subspaces In Exercises 720, W is not a subspac...
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 4.3.19: Subsets That Are Not Subspaces In Exercises 720, W is not a subspac...
 4.3.4.3.20: Subsets That Are Not Subspaces In Exercises 720, W is not a subspac...
 4.3.20: Subsets That Are Not Subspaces In Exercises 720, W is not a subspac...
 4.3.4.3.21: Determining Subspaces of C(, ) In Exercises 2128, determine whether...
 4.3.21: Determining Subspaces of C(, ) In Exercises 2128, determine whether...
 4.3.4.3.22: Determining Subspaces of C(, ) In Exercises 2128, determine whether...
 4.3.22: Determining Subspaces of C(, ) In Exercises 2128, determine whether...
 4.3.4.3.23: Determining Subspaces of C(, ) In Exercises 2128, determine whether...
 4.3.23: Determining Subspaces of C(, ) In Exercises 2128, determine whether...
 4.3.4.3.24: Determining Subspaces of C(, ) In Exercises 2128, determine whether...
 4.3.24: Determining Subspaces of C(, ) In Exercises 2128, determine whether...
 4.3.4.3.25: Determining Subspaces of C(, ) In Exercises 2128, determine whether...
 4.3.25: Determining Subspaces of C(, ) In Exercises 2128, determine whether...
 4.3.4.3.26: Determining Subspaces of C(, ) In Exercises 2128, determine whether...
 4.3.26: Determining Subspaces of C(, ) In Exercises 2128, determine whether...
 4.3.4.3.27: Determining Subspaces of C(, ) In Exercises 2128, determine whether...
 4.3.27: Determining Subspaces of C(, ) In Exercises 2128, determine whether...
 4.3.4.3.28: Determining Subspaces of C(, ) In Exercises 2128, determine whether...
 4.3.28: Determining Subspaces of C(, ) In Exercises 2128, determine whether...
 4.3.4.3.29: Determining Subspaces of Mn,n In Exercises 2936, determine whether ...
 4.3.29: Determining Subspaces of Mn,n In Exercises 2936, determine whether ...
 4.3.4.3.30: Determining Subspaces of Mn,n In Exercises 2936, determine whether ...
 4.3.30: Determining Subspaces of Mn,n In Exercises 2936, determine whether ...
 4.3.4.3.31: Determining Subspaces of Mn,n In Exercises 2936, determine whether ...
 4.3.31: Determining Subspaces of Mn,n In Exercises 2936, determine whether ...
 4.3.4.3.32: Determining Subspaces of Mn,n In Exercises 2936, determine whether ...
 4.3.32: Determining Subspaces of Mn,n In Exercises 2936, determine whether ...
 4.3.4.3.33: Determining Subspaces of Mn,n In Exercises 2936, determine whether ...
 4.3.33: Determining Subspaces of Mn,n In Exercises 2936, determine whether ...
 4.3.4.3.34: Determining Subspaces of Mn,n In Exercises 2936, determine whether ...
 4.3.34: Determining Subspaces of Mn,n In Exercises 2936, determine whether ...
 4.3.4.3.35: Determining Subspaces of Mn,n In Exercises 2936, determine whether ...
 4.3.35: Determining Subspaces of Mn,n In Exercises 2936, determine whether ...
 4.3.4.3.36: Determining Subspaces of Mn,n In Exercises 2936, determine whether ...
 4.3.36: Determining Subspaces of Mn,n In Exercises 2936, determine whether ...
 4.3.4.3.37: Determining Subspaces of R3 In Exercises 3742, determine whether th...
 4.3.37: Determining Subspaces of R3 In Exercises 3742, determine whether th...
 4.3.4.3.38: Determining Subspaces of R3 In Exercises 3742, determine whether th...
 4.3.38: Determining Subspaces of R3 In Exercises 3742, determine whether th...
 4.3.4.3.39: Determining Subspaces of R3 In Exercises 3742, determine whether th...
 4.3.39: Determining Subspaces of R3 In Exercises 3742, determine whether th...
 4.3.40: Determining Subspaces of R3 In Exercises 3742, determine whether th...
 4.3.4.3.40: Determining Subspaces of R3 In Exercises 3742, determine whether th...
 4.3.41: Determining Subspaces of R3 In Exercises 3742, determine whether th...
 4.3.4.3.41: Determining Subspaces of R3 In Exercises 3742, determine whether th...
 4.3.42: Determining Subspaces of R3 In Exercises 3742, determine whether th...
 4.3.4.3.42: Determining Subspaces of R3 In Exercises 3742, determine whether th...
 4.3.43: True or False? In Exercises 43 and 44, determine whether each state...
 4.3.4.3.43: True or False? In Exercises 43 and 44, determine whether each state...
 4.3.44: True or False? In Exercises 43 and 44, determine whether each state...
 4.3.4.3.44: True or False? In Exercises 43 and 44, determine whether each state...
 4.3.45: Consider the vector spacesP0, P1, P2, . . . , Pn where Pk is the se...
 4.3.4.3.45: Consider the vector spacesP0, P1, P2, . . . , Pn where Pk is the se...
 4.3.46: Calculus Let W1, W2, W3, W4, and W5 be defined as in Example 5. Sho...
 4.3.4.3.46: Calculus Let W1, W2, W3, W4, and W5 be defined as in Example 5. Sho...
 4.3.47: Calculus Let F(, ) be the vector space ofrealvalued functions defi...
 4.3.4.3.47: Calculus Let F(, ) be the vector space ofrealvalued functions defi...
 4.3.48: Calculus Determine whether the setS = {f C[0, 1]:10f(x) dx = 0}is a...
 4.3.4.3.48: Calculus Determine whether the setS = {f C[0, 1]:10f(x) dx = 0}is a...
 4.3.49: Let W be the subset of R3 consisting of all points on aline that pa...
 4.3.4.3.49: Let W be the subset of R3 consisting of all points on aline that pa...
 4.3.50: CAPSTONE Explain why it is sufficient totest for closure to establi...
 4.3.4.3.50: CAPSTONE Explain why it is sufficient totest for closure to establi...
 4.3.51: Guided Proof Prove that a nonempty set W is asubspace of a vector s...
 4.3.4.3.51: Guided Proof Prove that a nonempty set W is asubspace of a vector s...
 4.3.52: Let x, y, and z be vectors in a vector space V. Show thatthe set of...
 4.3.4.3.52: Let x, y, and z be vectors in a vector space V. Show thatthe set of...
 4.3.53: Proof Let A be a fixed 2 3 matrix. Prove that the setW = {x R3: Ax ...
 4.3.4.3.53: Proof Let A be a fixed 2 3 matrix. Prove that the setW = {x R3: Ax ...
 4.3.54: Proof Let A be a fixed m n matrix. Prove that the setW = {x Rn: Ax ...
 4.3.4.3.54: Proof Let A be a fixed m n matrix. Prove that the setW = {x Rn: Ax ...
 4.3.55: Proof Let W be a subspace of the vector space V. Prove that the zer...
 4.3.4.3.55: Proof Let W be a subspace of the vector space V. Prove that the zer...
 4.3.56: Give an example showing that the union of two subspaces of a vector...
 4.3.4.3.56: Give an example showing that the union of two subspaces of a vector...
 4.3.57: Proof Let A and B be fixed 2 2 matrices. Prove thatthe setW = {X: X...
 4.3.4.3.57: Proof Let A and B be fixed 2 2 matrices. Prove thatthe setW = {X: X...
 4.3.58: Proof Let V and W be two subspaces of a vectorspace U.(a) Prove tha...
 4.3.4.3.58: Proof Let V and W be two subspaces of a vectorspace U.(a) Prove tha...
 4.3.59: Proof Complete the proof of Theorem 4.6 by showingthat the intersec...
 4.3.4.3.59: Proof Complete the proof of Theorem 4.6 by showingthat the intersec...
Solutions for Chapter 4.3: Subspaces of Vector Spaces
Full solutions for Elementary Linear Algebra  8th Edition
ISBN: 9781305658004
Solutions for Chapter 4.3: Subspaces of Vector Spaces
Get Full SolutionsChapter 4.3: Subspaces of Vector Spaces includes 118 full stepbystep solutions. Since 118 problems in chapter 4.3: Subspaces of Vector Spaces have been answered, more than 44025 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 8. Elementary Linear Algebra was written by and is associated to the ISBN: 9781305658004.

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

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).

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.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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

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  ).

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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).

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

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

Solvable system Ax = b.
The right side b is in the column space of A.

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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