In each case, construct a matrix with the requisite properties or explain why no such

Find a basis for each of the given subspaces and determine its dimension. a. \(V=\operatorname{Span}((1,2,3),(3,4,7),(5,-2,3)) \subset \mathbb{R}^{3}\) b. \(V=\left\{\mathbf{x} \in \mathbb{R}^{4}: x_{1}+x_{2}+x_{3}+x_{4}=0, x_{2}+x_{4}=0\right\} \subset \mathbb{R}^{4}\) c. \(V=(\operatorname{Span}((1,2,3)))^{\perp} \subset \mathbb{R}^{3}\) d. \(V=\left\{\mathbf{x} \in \mathbb{R}^{5}: x_{1}=x_{2}, x_{3}=x_{4}\right\} \subset \mathbb{R}^{5}\)

In Example 3, we were given v1 = 1 1 1 1 , v2 = 0 1 3 1 R4 and constructed v3, v4 so that {v1, v2, v3, v4} gives a basis for R4. Here is an alternative construction: Consider the collection of vectors v1, v2, e1, e2, e3, e4; these certainly span R4. Now use the approach of Example 2 to find a basis {v1, v2, . . . }.

For each of the following matrices A, give bases for R(A), N(A), C(A), and N(AT). Check dimensions and orthogonality. a. A = _ 1 2 3 2 4 6 _ b. A = 2 1 3 4 3 5 3 3 3 c. A = _ 1 2 1 0 2 4 3 1 _ d. A = 1 1 1 1 0 1 0 2 1 1 0 2 2 2 0 1 1 1 0 1 e. A = 1 1 0 1 1 1 1 2 1 1 2 2 2 0 0 1 1 2 3 3 f. A = 1 1 0 5 0 1 0 1 1 3 2 0 1 2 3 4 1 6 0 4 4 12 1 7

Find a basis for the intersection of the subspaces V = Span _ (1, 0, 1, 1), (2, 1, 1, 2) _ and W = Span _ (0, 1, 1, 0), (2, 0, 1, 2) _ R4.

Give a basis for the orthogonal complement of each of the following subspaces of R4. a. V = Span _ (1, 0, 3, 4), (0, 1, 2,5) _ b. W = {x R4 : x1 + 3x3 + 4x4 = 0, x2 + 2x3 5x4 = 0}

a. Give a basis for the orthogonal complement of the subspace V R4 given by V = {x R4 : x1 + x2 2x4 = 0, x1 x2 x3 + 6x4 = 0, x2 + x3 4x4 = 0}. b. Give a basis for the orthogonal complement of the subspace W R4 spanned by (1, 1, 0,2), (1,1,1, 6), and (0, 1, 1,4). c. Give a matrix B so that the subspace W defined in part b can be written in the form W = N(B).

We saw in Exercise 2.5.13 that if the \(m \times n\) matrix A has rank 1, then there are nonzero vectors \(\mathbf{u} \in \mathbb{R}^{m} \text { and } \mathbf{v} \in \mathbb{R}^{n}\) such that \(A=\mathbf{u v}^{\top}\). Describe the four fundamental subspaces of A in terms of u and v. (Hint: What are the columns of \(\mathbf{u v}^{\mathrm{T}}\)?)

In each case, construct a matrix with the requisite properties or explain why no such matrix exists. a. The column space has basis 1 0 1 , and the nullspace contains 1 2 0 . b. The nullspace contains 1 0 1 , 1 2 1 , and the row space contains 1 1 1 . The column space has basis 1 0 1 , 0 1 1 , and the row space has basis 1 1 1 , 2 0 1 . d. The column space and the nullspace both have basis _ 1 0 _ . e. The column space and the nullspace both have basis 1 0 0 .

Given the LU decompositions of the following matrices A, give bases for \(\mathbf{R}(A), \mathbf{N}(A)\), \(\mathbf{C}(A)\), and \(\mathbf{N}\left(A^{\top}\right)\). (Do not multiply out!) Check dimensions and orthogonality. a. \(A=\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right]\left[\begin{array}{rrr}2 & 2 & 4 \\ 0 & 1 & -1\end{array}\right]\) b. \(A=\left[\begin{array}{rrr}1 & 0 & 0 \\ 1 & 1 & 0 \\ -2 & 0 & 1\end{array}\right]\left[\begin{array}{rrrr}1 & 1 & 2 & 1 \\ 0 & 0 & 2 & -1 \\ 0 & 0 & 0 & 0\end{array}\right]\) c. \(A=\left[\begin{array}{rrrr}1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & 2 & 1 & 0 \\ 3 & 0 & 1 & 1\end{array}\right]\left[\begin{array}{rrr}2 & 4 & -2 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]\)

According to Proposition 4.10, if A is an \(m \times n\) matrix, then for each \(\mathbf{b} \in \mathbf{C}(\Lambda)\), there is a unique \(\mathbf{x} \in \mathbf{R}(\Lambda)\) with \(A \mathbf{x}=\mathbf{b}\). In each case, give a formula for that \(\mathbf{x}\). a. \(A=\left[\begin{array}{lll}1 & 2 & 3 \\ 1 & 2 & 3\end{array}\right]\) b. \(A=\left[\begin{array}{rrr}1 & 1 & 1 \\ 0 & 1 & -1\end{array}\right]\)

Let \(A=\left[\begin{array}{rrrr}1 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1\end{array}\right]\). a. Given any \(\mathbf{x} \in \mathbb{R}^4\), find \(\mathbf{u} \in \mathbf{R}(A)\) and \(\mathbf{v} \in \mathbf{N}(A)\) so that \(\mathbf{x}=\mathbf{u}+\mathbf{v}\). b. Given \(\mathbf{b} \in \mathbb{R}^2\), give the unique element \(\mathbf{x} \in \mathbf{R}(A)\) so that \(A \mathbf{x}=\mathbf{b}\).

Let A be an n n matrix. Prove that A is singular if and only if AT is singular.

Let A be an m n matrix with rank r. Suppose A = BU, where U is in echelon form. Show that the first r columns of B give a basis for C(A). (In particular, if EA = U, where U is the echelon form of A and E is the product of elementary matrices by which we reduce A to U, then the first r columns of E 1 give a basis for C(A).)

Recall from Exercise 3.1.13 that for any subspace V Rn we have V (V ) . Give alternative proofs of Proposition 3.6 a. by applying Proposition 4.3; b. by applying Theorem 4.9 to prove that if x (V ) , then x V .

Let \(V \subset \mathbb{R}^n\) be a subspace, let \(\left\{\mathbf{v}_1, \ldots, \mathbf{v}_k\right\}\) be a basis for V, and let \(\mathbf{w}_1, \ldots, \mathbf{w}_{\ell} \in V\) be vectors such that S pan \(\left(\mathbf{w}_1, \ldots, \mathbf{w}_{\ell}\right)=V\). Prove that \(\ell \geq k\).

Prove Proposition 4.4. (Hint: Exercise 15 and Proposition 4.3 may be useful.) _

Let \(V \subset \mathbb{R}^{n}\) be a subspace, and suppose you are given a linearly independent set of vectors \(\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{k}\right\} \subset V\) . Show that if dim V > k, then there are vectors \(\mathbf{v}_{k+1}, \ldots, \mathbf{v}_{\ell} \in V\) so that \left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{\ell}\right\}\ forms a basis for V.

Suppose V andW are subspaces ofRn andW V . Prove that dimW dim V . (Hint: Start with a basis for W and apply Exercise 17.)

Suppose A is an \(n \times n\) matrix, and let \(\mathbf{v}_{1}, \ldots, \mathbf{v}_{n} \in \mathbb{R}^{n}\). Suppose \(\left\{A \mathbf{v}_{1}, \ldots, A \mathbf{v}_{n}\right\}\) is linearly independent. Prove that A is nonsingular.

Continuing Exercise 3.3.23: Let U and V be subspaces of \(\mathbb{R}^n\). Prove that if \(U \cap V= \{\mathbf 0\}\), then \(\operatorname{dim}(U+V)=\operatorname{dim} U+\operatorname{dim} V\).

Let U and V be subspaces of Rn. Prove that dim(U + V ) = dim U + dim V dim(U V ). (Hint: This is a generalization of Exercise 20. Start with a basis for U V , and use Exercise 17.)

Continuing Exercise 3.2.10: Let A be an m n matrix, and let B be an n p matrix. a. Prove that rank(AB) rank(A). (Hint: Look at part b of Exercise 3.2.10.) b. Prove that if n = p and B is nonsingular, then rank(AB) = rank(A). c. Prove that rank(AB) rank(B). (Hint: Use part a of Exercise 3.2.10 and Theorem 4.6.) d. Prove that if m = n and A is nonsingular, then rank(AB) = rank(B). e. Prove that if rank(AB) = n, then rank(A) = rank(B) = n.

a. Let A be an \(m \times n\) matrix, and let B be an \(n \times p\) matrix. Show that \(A B=\mathrm{O} \Longleftrightarrow \mathbf{C}(B) \subset \mathbf{N}(A)\). b. Suppose A and B are \(3 \times 3\) matrices of rank 2. Show that \(A B \neq 0\). c. Give examples of \(3 \times 3\) matrices A and B of rank 2 so that AB has each possible rank.

Continuing Exercise 3.2.10: Let A be an \(m \times n\) matrix. a. Use Theorem 2.5 to prove that \(\mathbf{N}\left(A^{\top} A\right)=\mathbf{N}(A)\). (Hint: If \(\mathbf{x} \in \mathbf{N}\left(A^{\top} A\right)\), then \(A \mathbf{x} \in \mathbf{C}(A) \cap \mathbf{N}\left(A^{\top}\right)\).) b. Prove that \(\operatorname{rank}(A)=\operatorname{rank}\left(A^{\top} A\right)\). c. Prove that \(\mathbf{C}\left(A^{\top} A\right)=\mathbf{C}\left(A^{\top}\right)\).

In this exercise, we investigate the composition of functions AT A mapping R(A) to R(A), pursuing the discussion on p. 167. a. Suppose A is an m n matrix. Show that ATA = In if and only if the column vectors a1, . . . , an Rm are mutually orthogonal unit vectors. b. Suppose A is anm n matrix of rank 1. Using the notation of Exercise 7, show that ATAx = x for each x R(A) if and only if _u__v_ = 1. Use this fact to show that we can write A = uvT, where _u_ = _v_ = 1. Interpret A geometrically. (See Exercise 4.4.22 for a generalization.)

Generalizing Exercise 3.2.13: Suppose A is an n n matrix with the property that rank(A) = rank(A2). a. Show that N(A2) = N(A). b. Prove that C(A) N(A) = {0}. c. Conclude that C(A) + N(A) = Rn. (Hint: Use Exercise 20.)