Given each matrix A, find matrices X and Y so that C(A) = N(X) and N(A) = C(Y ). a. A =

Show that if B is obtained from A by performing one or more row operations, then R(B) = R(A).

What vectors b are in the column space of A in each case? (Give constraint equations.) Check that the coefficients of the constraint equations give linear combinations of the rows of A summing to 0. a. A = 3 1 6 2 9 3 b. A = 1 1 1 1 1 2 1 3 5 c. A = 1 2 1 0 1 3 1 3 0 1 0 1

Given each matrix A, find matrices X and Y so that C(A) = N(X) and N(A) = C(Y ). a. A = 3 1 6 2 9 3 b. A = 1 1 0 2 1 1 1 1 2 c. A = 1 1 1 1 2 0 1 1 1 1 0 2

Let A = 1 2 1 1 1 0 3 4 2 2 2 3 . a. Give constraint equations for C(A). b. Find vectors spanning N(AT).

Let \(L=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 1 & -1 & 1 \end{array}\right] \text { and } U=\left[\begin{array}{cccr} 1 & 2 & 1 & 1 \\ 0 & 0 & 2 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right] \text {. }\) If A = LU, give vectors that span R(A), C(A), and N(A).

a. Construct a matrix whose column space contains 1 1 1 and 0 1 1 and whose nullspace contains 1 0 1 and 0 1 0 , or explain why none can exist. b. Construct a matrix whose column space contains 1 1 1 and 0 1 1 and whose nullspace contains 1 0 1 0 and 1 0 0 1 , or explain why none can exist.

Let \(A=\left[\begin{array}{lll}1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1\end{array}\right]\) and \(B=\left[\begin{array}{rrr}1 & 1 & 0 \\ -1 & -1 & 0 \\ 0 & -1 & -1\end{array}\right]\). a. Give \(\mathbf{C}(A)\) and \(\mathbf{C}(B)\). Are they lines, planes, or all of \(\mathbb{R}^3\) ? b. Describe \(\mathbf{C}(A+B)\) and \(\mathbf{C}(A)+\mathbf{C}(B)\). Compare your answers.

a. Construct a 3 3 matrix A with C(A) N(A). b. Construct a 3 3 matrix A with N(A) C(A). c. Do you think there can be a 3 3 matrix A with N(A) = C(A)? Why or why not? d. Construct a 4 4 matrix A with C(A) = N(A).

Let A be an \(m \times n\) matrix and recall that we have the associated function \(\mu_A: \mathbb{R}^n \rightarrow\) \(\mathbb{R}^m\) defined by \(\mu_A(\mathbf{x})=A \mathbf{x}\). Show that \(\mu_A\) is a one-to-one function if and only if \(\mathbf{N}(A)=\{0\}\)

Let A be an \(m \times n\) matrix and B be an \(n \times p\) matrix. Prove that a. \(\mathbf{N}(B) \subset \mathbf{N}(A B)\). b. \(\mathbf{C}(A B) \subset \mathbf{C}(A)\). (Hint: Use Proposition 2.1.) c. \(\mathbf{N}(B)=\mathbf{N}(A B)\) when A is \(n \times n\) and nonsingular. (Hint: See the box on p. 12.) d. \(\mathbf{C}(A B)=\mathbf{C}(A)\) when B is \(n \times n\) and nonsingular.

Let A be an m n matrix. Prove that N(ATA) = N(A). (Hint: Use Exercise 10 and Exercise 2.5.15.)

Suppose A and B are m n matrices. Prove that C(A) and C(B) are orthogonal subspaces of Rm if and only if ATB = O.

Suppose A is an \(n \times n\) matrix with the property that \(A^2=A\). a. Prove that \(\mathbf{C}(A)=\left\{\mathbf{x} \in \mathbb{R}^n: \mathbf{x}=A \mathbf{x}\right\}\). b. Prove that \(\mathbf{N}(\Lambda)=\left\{\mathbf{x} \in \mathbb{R}^n: \mathbf{x}=\mathbf{u}-\Lambda \mathbf{u}\right.\) for some \(\left.\mathbf{u} \in \mathbb{R}^n\right\}\). c. Prove that \(\mathbf{C}(A) \cap \mathbf{N}(A)=\{0\}\). d. Prove that \(\mathbf{C}(A)+\mathbf{N}(A)=\mathbb{R}^n\).