3.4Nullspace and left nullspace

What is nullspace

• The nullspace of A, donoted Null(A),is the solution set of $A\vec{x}=\vec{0}$

• the left nullspace of A is Null($A^T$) (solving $A^T\vec{x}=\vec{0}$)

Nullity

• The dimension of Row(A) and Col(A) is the same as the rank of A

• The Nullity OF A is the dimension of Null(A)

  • Rank - Nulity theorem

    • If A is an mxn matrix , then rank (A)+nulity(A)=n

      • (nulity is the number of free variables)

Null(A) is “orthogonal” to row(A) in this sense:

$$Null(A)=[\vec{n}\in {\mathbb{R}}^n|\vec{r}\cdot \vec{n}=0,\vec{r}\in Row(A)]$$

• EX.

  • use the nullspace to find the set of all vectors $\vec{x}\in {\mathbb{R}}^4$ orthoganal to both ${\vec{v}}_1=(1,0,-1,1)$ and ${\vec{v}}_2=(1,1,1,2)$ (put ${\vec{v}}_1,{\vec{v}}_2$ as rows of A)

$$\begin{array}{r}A=\left[\begin{array}{c}<-v_1->\\ <-v_2->\end{array}\right]=\left[\begin{array}{cccc}1& 0& -1& 1\\ 1& 1& 1& 2\end{array}\right]\\ RREF=\left[\begin{array}{ccccc}1& 0& -1& 1& |0\\ 0& 1& 2& 1& |0\end{array}\right](x_3,x_4⟹free)\\ let,x_3=s,x_4=t⟹x_2=-2s-t,x_1=s-t\\ \left[\begin{array}{c}x1\\ x2\\ x3\\ x4\end{array}\right]=\left[\begin{array}{c}s-t\\ -2s-t\\ s\\ t\end{array}\right]\end{array}$$