What is the kernel of this matrix?

Question

Consider the matrix \begin{bmatrix}0&1&0\\0&0&1\\0&0&0\end{bmatrix} Why is the kernel equal to \begin{bmatrix}0\\0\\0\end{bmatrix} rather than \begin{bmatrix}1\\0\\0\end{bmatrix}?

Answer 1

The kernel is $$\{t\cdot \begin{bmatrix} 1\\0\\0\end{bmatrix}:t\in \mathbb R\}$$...

It's easy to see that that this set is contained in the kernel (by matrix multiplication)...

By the Rank-nullity theorem the kernel is $1$-dimensional (the rank is $2$)...

Answer 2

The matrix is row-reduced. You can read off the solutions to $A\vec x= \vec 0$ and see that the kernel consists of the vectors $\begin{bmatrix}x\\y\\z\end{bmatrix}$ with $y=z=0$...