4.3 Row and Column Spaces

1. Find bases for the row, column and null spaces of the following matrices. Verify the Rank-Nullity Theorem. a. $A=\left[\begin{array}{ccc}1& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right]$ Row Basis: $[1,0,1],[0,1,1]$ (found by REF) Column Basis: $[1,2,1],[0,1,-1]$ $x_1+x_3=0$ $x_2+x_3=0$ Let $x_3=t$, and t is a real number $x_1=-t$ $x_2=-t$ $x_3=t$ Null Space Basis: $[-1,-1,1]$

$\text{rank}(A)+\text{nullity}(A)=n$ $2+1=3$ Rank Nullity Theorem Holds True

b. $A=\left[\begin{array}{cccc}-1& 4& 3& 0\\ 2& -4& -4& 2\\ 2& 8& 2& 8\end{array}\right]$ $\text{rref}(A)=\left[\begin{array}{cccc}1& 0& -1& 2\\ 0& 1& \frac{1}{2}& \frac{1}{2}\\ 0& 0& 0& 0\end{array}\right]$ Row Basis: $[1,0,-1,2],[0,1,\frac{1}{2},\frac{1}{2}]$ Column Basis: $[-1,2,2],[4,-4,8]$ $x_1-x_3+2x_4=0$ $x_2+\frac{1}{2}{x}_3+\frac{1}{2}{x}_4=0$ Let $x_3=s$, $x_4=t$ $x_1=s-2t$ $x_2=-\frac{1}{2}s-\frac{1}{2}t$ $x_3=s$ $x_4=t$ Null Space Basis: $[1,-\frac{1}{2},1,0],[-2,-\frac{1}{2},0,1]$

1. Find all values of $h$ such that $\mathrm{rank}(A)=2$

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

3. Prove that if $A\in {\mathbb{R}}^{n\times m}$ then $\mathrm{rank}(A)=\mathrm{rank}\left(A^T\right)$