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## Homework Statement

Find a kernel and image basis of the linear transformation having:

[tex] \displaystyle T:{{\mathbb{R}}^{3}}\to {{\mathbb{R}}^{3}}[/tex] so that

[tex] \displaystyle _{B}{{\left( T \right)}_{B}}=\left( \begin{matrix}

1 & 2 & 1 \\

2 & 4 & 2 \\

0 & 0 & 0 \\

\end{matrix} \right)[/tex]

[tex] \displaystyle B=\left\{ \left( 1,1,0 \right),\left( 0,2,0 \right),\left( 2,0,-1 \right) \right\}[/tex]

## Homework Equations

## The Attempt at a Solution

For the image basis it is easy given the fact that the rank of the associated matrix is 1 so, the image is generated by one column.

The problem comes when finding the Kernel basis. My idea is to save the general fromula of the linear map which would work for sure but I wanted to know if there's a quicker way of doing it without finding the general formula of the linear map.

Thanks!