Linear Algebra with Applications, 10th Edition

Section 5.2 Exercises

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

2. 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 $S^{\perp }$.

3. 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 ${\left(1,-1,2\right)}^T$.

4. 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 $S^{\perp }$.

5. 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 ...