For each of the following matrices, find a basis for the row space, a basis for the column space, and a basis for the null space:

1. $\left[\begin{array}{ccc}1& 3& 2\\ 2& 1& 4\\ 4& 7& 8\end{array}\right]$

2. $\left[\begin{array}{cccc}\hfill -3& 1& \hfill 3& \hfill 4\\ \hfill 1& 2& \hfill -1& \hfill -2\\ \hfill -3& 8& \hfill 4& \hfill 2\end{array}\right]$

3. $\left[\begin{array}{cccc}\hfill 1& 3& \hfill -2& \hfill 1\\ \hfill 2& 1& \hfill 3& \hfill 2\\ \hfill 3& 4& \hfill 5& \hfill 6\end{array}\right]$

In each of the following, determine the dimension of the subspace of ${ℝ}^3$ spanned by the given vectors:

1. $\left[\begin{array}{r}1\\ -2\\ 2\end{array}\right],\left[\begin{array}{r}2\\ -2\\ 4\end{array}\right],\left[\begin{array}{r}-3\\ 3\\ 6\end{array}\right]$

2. $\left[\begin{array}{r}1\\ 1\\ 1\end{array}\right],\left[\begin{array}{r}1\\ 2\\ 3\end{array}\right],\left[\begin{array}{r}2\\ 3\\ 1\end{array}\right]$

3. $\left[\begin{array}{r}1\\ -1\\ 2\end{array}\right],\left[\begin{array}{r}-2\\ 2\\ -4\end{array}\right],\left[\begin{array}{r}3\\ -2\\ 5\end{array}\right],\left[\begin{array}{r}2\\ -1\\ 3\end{array}\right]$

Let

1. Compute the reduced row echelon form U of A. Which column vectors of U correspond to the free variables? Write each of these vectors as a linear combination of the column vectors corresponding to the lead variables.

2. Which column vectors of A correspond to the lead variables of U? These column vectors form a basis for the column space of A. Write each of the remaining column vectors of A as a linear combination of these basis vectors.

For each of the following choices of A and b, determine whether b is in the column space of A and state whether the system $A\mathbf{x}=\mathbf{b}$ is consistent:

1. $\begin{array}{cc}A=\left[\begin{array}{cc}1& 2\\ 2& 4\end{array}\right],& \mathbf{b}=\left[\begin{array}{c}4\\ 8\end{array}\right]\end{array}$

2. $\begin{array}{cc}A=\left[\begin{array}{cc}3& 6\\ 1& 2\end{array}\right],& \mathbf{b}=\left[\begin{array}{c}1\\ 1\end{array}\right]\end{array}$

3. $\begin{array}{cc}A=\left[\begin{array}{cc}2& 1\\ 3& 4\end{array}\right],& \mathbf{\text{b}}=\left[\begin{array}{c}4\\ 6\end{array}\right]\end{array}$

4. $\begin{array}{cc}A=\left[\begin{array}{ccc}1& 1& 2\\ 1& 1& 2\\ 1& 1& 2\end{array}\right],& \mathbf{b}=\left[\begin{array}{c}\begin{array}{c}1\\ 2\end{array}\\ 3\end{array}\right]\end{array}$

5. $\begin{array}{cc}A=\left[\begin{array}{cc}0& 1\\ 1& 0\\ 0& 1\end{array}\right],& \mathbf{b}=\left[\begin{array}{c}\begin{array}{c}2\\ 5\end{array}\\ 2\end{array}\right]\end{array}$

6. $\begin{array}{cc}A=\left[\begin{array}{cc}1& 2\\ 2& 4\\ 1& 2\end{array}\right],& \mathbf{b}=\left[\begin{array}{r}5\\ 10\\ 5\end{array}\right]\end{array}$

For each consistent system in Exercise 4, determine whether there will be one or infinitely many solutions by examining the column vectors of the coefficient matrix A.

How many solutions will the linear system $A\mathbf{x}=\mathbf{b}$ have if b is in the column space of A and the column vectors of A are linearly dependent? Explain.

Let A be a $6\times n$ matrix of rank r and let b beavectorin ${ℝ}^6$. For each choice of r and n that follows, indicate the possibilities as to the number of solutions one could have for the linear system $A\mathbf{x}=\mathbf{b}$. Explain your answers.

1. $n=7,r=5$

2. $n=7,r=6$

3. $n=5,r=5$

4. $n=5,r=4$

Let A be an $m\times n$ matrix with $m>n$. Let $\mathbf{b}\in {ℝ}^m$ and suppose that $N\left(A\right)=\left\{\mathbf{0}\right\}$.

1. What can you conclude about the column vectors of A? Are they linearly independent? Do they span ${ℝ}^m$? Explain.

2. How many solutions will the system $A\mathbf{x}=\mathbf{b}$ have if b is not in the column space of A? How many solutions will there be if b is in the column space of A? Explain.

Let A and B be $6\times 5$ matrices. If dim $N\left(A\right)=2$, what is the rank of A? If the rank of B is 4, what is the dimension of $N\left(B\right)$?

Let A be an $m\times n$ matrix whose rank is equal to n. If $A\mathbf{c}=A\mathbf{d}$, does this imply that c must be equal to d? What if the rank of A is less than n? Explain your answers.

Let A be an $m\times n$ matrix. Prove that

Let A and B be row equivalent matrices.

1. Show that the dimension of the column space of A equals the dimension of the column space of B.

2. Are the column spaces of the two matrices necessarily the same? Justify your answer.

Let A be a $4\times 3$ matrix and suppose that the vectors

form a basis for $N\left(A\right)$. If $\mathbf{b}={\mathbf{a}}_1+2{\mathbf{a}}_2+{\mathbf{a}}_3$, find all solutions of the system $A\mathbf{x}=\mathbf{b}$.

Let A be a 4 × 4 matrix with reduced row echelon form given by

If

find ${\mathbf{a}}_3$ and ${\mathbf{a}}_4$.

Let A be a $4\times 5$ matrix and let U be the reduced row echelon form of A. If

1. find a basis for $N\left(A\right)$.

2. given that ${\mathbf{x}}_0$ is a solution to $A\mathbf{x}=\mathbf{b}$, where

   $$\begin{array}{ccc}\mathbf{b}=\left[\begin{array}{c}0\\ 5\\ 3\\ 4\end{array}\right]& \text{and}& {\mathbf{x}}_0=\left[\begin{array}{c}3\\ 2\\ 0\\ 2\\ 0\end{array}\right]\end{array}$$

   1. find all solutions to the system.

   2. determine the remaining column vectors of A.

Let A be a $5\times 8$ matrix with rank equal to 5 and let b be any vector in ${ℝ}^5$. Explain why the system $A\mathbf{x}=\mathbf{b}$ must have infinitely many solutions.

Let A be a $4\times 5$ matrix. If ${\mathbf{a}}_1$, ${\mathbf{a}}_2$, and ${\mathbf{a}}_4$ are linearly independent and

determine the reduced row echelon form of A.

Let A be a $5\times 3$ matrix of rank 3 and let {x₁, ${\mathbf{x}}_2$, ${\mathbf{x}}_3$} be abasis for ${ℝ}^3$.

1. Show that $N\left(A\right)=\left\{\mathbf{0}\right\}$.

2. Show that if ${\text{y}}_1=A{\text{x}}_1,{\text{y}}_2=A{\text{x}}_2$, and ${\text{y}}_3=A{\text{x}}_3$, then ${\mathbf{y}}_1$, ${\mathbf{y}}_2$, and ${\mathbf{y}}_3$ are linearly independent.

3. Do the vectors ${\mathbf{y}}_1$, ${\mathbf{y}}_2$, ${\mathbf{y}}_3$ from part (b) form a basis for ${ℝ}^5$? Explain.

Let A be an $m\times n$ matrix with rank equal to n. Show that if $\mathbf{x}\ne \mathbf{0}$ and $\mathbf{y}=A\mathbf{x}$, then $\mathbf{y}\ne \mathbf{0}$.

Prove that a linear system $A\mathbf{x}=\mathbf{b}$ is consistent if and only if the rank of (A | b) equals the rank of A.

Let A and B be $m\times n$ matrices. Show that

Let A be an $m\times n$ matrix.

1. Show that if B is a nonsingular $m\times m$ matrix, then BA and A have the same null space and hence the same rank.

2. Show that if C is a nonsingular $n\times n$ matrix, then AC and A have the same rank.

Prove Corollary 3.6.4.

Show that if A and B are $n\times n$ matrices and $N(A-B)={ℝ}^n$, then $A=B$.

Let A and B be $n\times n$ matrices.

1. Show that $AB=O$ if and only if the column space of B is a subspace of the null space of A.

2. Show that if $AB=O$, then the sum of the ranks of A and B cannot exceed n.

Let A $A\in {ℝ}^{m\times n}$ and $\text{b}\in {ℝ}^m$, and let ${\mathbf{x}}_0$ be a particular solution of the system $A\mathbf{x}=\mathbf{b}$. Prove that if $N\left(A\right)=\left\{\mathbf{0}\right\}$, then the solution ${\mathbf{x}}_0$ must be unique.

Let x and y be nonzero vectors in ${ℝ}^m$ and ${ℝ}^n$, respectively, and let $A={\mathbf{\text{xy}}}^T$.

1. Show that {x} is a basis for the column space of A and that $\left\{{\mathbf{y}}^T\right\}$ is a basis for the row space of A.

2. What is the dimension of $N\left(A\right)$?

Let $A\in {ℝ}^{m\times n}$, $B\in {ℝ}^{m\times r}$, and $C=AB$. Show that

1. the column space of C is a subspace of the column space of A.

2. the row space of C is a subspace of the row space of B.

3. $\text{rank}\left(C\right)\le \text{min}\{\text{rank}\left(A\right),\text{rank}\left(B\right)\}$.

Let $A\in {ℝ}^{m\times n}$, $B\in {ℝ}^{n\times r}$, and $C=AB$. Show that

1. if A and B both have linearly independent column vectors, then the column vectors of C will also be linearly independent.

2. if A and B both have linearly independent row vectors, then the row vectors of C will also be linearly independent. [Hint: Apply part (a) to $C^T$.]

Let $A\in {ℝ}^{m\times n}$, $B\in {ℝ}^{n\times r}$, and $C=AB$. Show that

1. if the column vectors of B are linearly dependent, then the column vectors of C must be linearly dependent.

2. if the row vectors of A are linearly dependent, then the row vectors of C are linearly dependent. [Hint: Apply part (a) to $C^T$.]

An $m\times n$ matrix A is said to have a right inverse if there exists an $n\times m$ matrix C such that $AC=I_m$. The matrix A is said to have a left inverse if there exists an $n\times m$ matrix D such that $DA=I_n$.

1. Show that if A has a right inverse, then the column vectors of A span ${ℝ}^m$.

2. Is it possible for an $m\times n$ matrix to have a right inverse if $n<m?n\ge m?$ Explain.

Prove: If A is an $m\times n$ matrix and the column vectors of A span ${ℝ}^m$, then A has a right inverse. [Hint: Let ${\mathbf{e}}_j$ denote the jth column of $I_m$ and solve $A\mathbf{x}={\mathbf{e}}_j$ for $j=1,\dots ,m$.]

Show that a matrix B has a left inverse if and only if $B^T$ has a right inverse.

Let B be an $n\times m$ matrix whose columns are linearly independent. Show that B has a left inverse.

Prove that if a matrix B has a left inverse, then the columns of B are linearly independent.

Show that if a matrix U is in row echelon form, then the nonzero row vectors of U form a basis for the row space of U.