For each of the following matrices, determine a basis for each of the subspaces $R\left(A^T\right)$, N(A), R(A), and N($A^T$).

1. $A=\left[\begin{array}{cc}3& 4\\ 6& 8\end{array}\right]$

2. $A=\left[\begin{array}{ccc}1& 3& 1\\ 2& 4& 0\end{array}\right]$

3. $A=\left[\begin{array}{cc}4& -2\\ 1& 3\\ 2& 1\\ 3& 4\end{array}\right]$

4. $A=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 1& 1\\ 0& 0& 1& 1\\ 1& 1& 2& 2\end{array}\right]$

Let S be the subspace of ${ℝ}^3$ spanned by $\mathbf{x}={\left(1,-1,1\right)}^T$.

1. Find a basis for $S^{\perp }$.

2. Give a geometrical description of S and $S^{\perp }$.

1. Let S be the subspace of ${ℝ}^3$ spanned by the vectors $\mathbf{x}={\left(x_1,x_2,x_3\right)}^T$ and $\mathbf{y}={\left(y_1,y_2,y_3\right)}^T$. Let

   $$A=\left[\begin{array}{ccc}{x}_1& x_2& x_3\\ y_1& y_2& y_3\end{array}\right]$$

   Show that $S^{\perp }=N\left(A\right)$.

2. Find the orthogonal complement of the subspace of ${ℝ}^3$ spanned by ${\left(1,2,1\right)}^T$ and ${\left(1,-1,2\right)}^T$.

Let S be the subspace of ${ℝ}^4$ spanned by ${\mathbf{x}}_1={\left(1,0,-2,1\right)}^T$ and ${\mathbf{x}}_2={\left(0,1,3,-2\right)}^T$. Find a basis for $S^{\perp }$.

Let A be a $3\times 2$ matrix with rank 2. Give geometric descriptions of R(A) and $N\left(A^T\right)$, and describe geometrically how the subspaces are related.

Is it possible for a matrix to have the vector (3, 1, 2) in its row space and ${\left(2,1,1\right)}^T$ in its null space? Explain.

Let ${\mathbf{a}}_j$ be a nonzero column vector of an $m\times n$ matrix A. Is it possible for ${\mathbf{a}}_j$ to be in $N\left(A^T\right)$? Explain.

Let S be the subspace of ${ℝ}^n$ spanned by the vectors ${\mathbf{x}}_1,{\mathbf{x}}_2,\mathrm{\dots },{\mathbf{x}}_k$. Show that $\mathbf{y}\in S^{\perp }$ if and only if $\mathbf{y}\perp {\mathbf{x}}_i$ for $i=1,\mathrm{\dots },k$.

If A is an $m\times n$ matrix of rank r, what are the dimensions of N(A) and $N\left(A^T\right)$? Explain.

Prove Corollary 5.2.5.

Prove: If A is an $m\times n$ matrix and $\mathbf{x}\in {ℝ}^n$, then either $A\mathbf{x}=0$ or there exists $\mathbf{y}\in R\left(A^T\right)$ such that ${\mathbf{x}}^T\mathbf{y}\ne 0$. Draw a picture similar to Figure 5.2.2 to illustrate this result geometrically for the case where N(A) is a two-dimensional subspace of ${ℝ}^3$.

Let A be an $m\times n$ matrix. Explain why the following are true.

1. Any vector x in ${ℝ}^n$ can be uniquely written as a sum $\mathbf{y}+\mathbf{z}$, where $\mathbf{y}\in N\left(A\right)$ and $\mathbf{z}\in R\left(A^T\right)$.

2. Any vector $\mathbf{b}\in {ℝ}^m$ can be uniquely written as a sum $\mathbf{u}+\mathbf{v}$, where $\mathbf{u}\in N\left(A^T\right)$ and $\mathbf{v}\in R\left(A\right)$.

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

1. if $\mathbf{x}\in N\left(A^{T}A\right)$, then Ax is in both R(A) and $N\left(A^T\right)$.

2. $N\left(A^{T}A\right)=N\left(A\right)$.

3. A and $A^{T}A$ have the same rank.

4. if A has linearly independent columns, then $A^{T}A$ is nonsingular.

Let A be an $m\times n$ matrix, B an $n\times r$ matrix, and $C=AB$. Show that

1. N(B) is a subspace of N(C).

2. $N{\left(C\right)}^{\perp }$ is a subspace of $N{\left(B\right)}^{\perp }$ and, consequently, $R{\left(C\right)}^{\perp }$ is a subspace of $R{\left(B\right)}^{\perp }$.

Let U and V be subspaces of a vector space W. Show that if $W=U\oplus V$, then $U\cap V=\left\{0\right\}$.

Let A be an $m\times n$ matrix of rank r and let $\left\{{\mathbf{x}}_1,\mathrm{\dots },{\mathbf{x}}_r\right\}$ be a basis for $R\left(A^T\right)$. Show that $\left\{A{\mathbf{x}}_1,\mathrm{\dots },A{\mathbf{x}}_r\right\}$ is a basis for R(A).

Let x and y be linearly independent vectors in ${ℝ}^n$ and let $S=\text{Span}\left(\mathbf{x},\mathbf{y}\right)$. We can use x and y to define a matrix A by setting

1. Show that A is symmetric.

2. Show that $N\left(A\right)=S^{\perp }$.

3. Show that the rank of A must be 2.