Question

Find a basis for the null space, row space and column space of matrix

Answer 1

Let the matrix is $A=\begin{bmatrix} 1 & 0 & 0 & 1\\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \end{bmatrix}$

Basis for row space of $A=\begin{Bmatrix} [1 & 0 & 0 & 1], [0 & 1 & 0 & 1],[0 & 0 & 1 & 1] \end{Bmatrix}$

In the 1, 2 and 3, it contains pivot hence,

$=\{\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\}$

If $Ax=0$

$\begin{bmatrix} x_1\\ x_2 \\ x_3\\ x_4 \end{bmatrix}$ $=\begin{bmatrix} -1\\ -1 \\ -1\\ 1 \end{bmatrix}$

Hence, it is much better for the null space $=\begin{bmatrix} -1\\ -1 \\ -1\\ 1 \end{bmatrix}$