Construct a 4 x 4 Matrix A such that {(1,2,3,4),(1,1,2,2)} is a basis of the N(A)
My confusion here lies in the fact that I need to build out a matrix given the basis of the null space. How can I use the definition of the null space of (A) to help me reconstruct the Matrix? I just don't 'get it'
Best Answer
You can take two vectors $v$ and $w$ such that $\{(1,2,3,4),(1,1,2,2),v,w\}$. For instance, you can take $v=(1,0,0,0)$ and $w=(0,1,0,0)$. Now consider the matrix$$M=\begin{bmatrix}1&1&1&0\\2&1&0&1\\3&2&0&0\\4&2&0&0\end{bmatrix},$$the columns of which are those four vectors. Finally, let$$B=\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}.$$Clearly, $\{(1,0,0,0),(0,1,0,0)\}$ is a basis of the null space of $B$. So, take\begin{align}A&=M.B.M^{-1}\\&=\begin{bmatrix}1 & 0 & -1 & \frac12 \\ 0 & 1 & 0 & -\frac12 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{bmatrix}.\end{align}