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 $\mathrm{rank}(A)=\mathrm{rank}\left(A^T\right)$