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- Apr 14, 2013

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Let $1\leq n,m\in \mathbb{N}$, $V:=\mathbb{R}^n$ and $(b_1, \ldots , b_n)$ a basis of $V$. Let $W:=\mathbb{R}^m$ and let $\phi:V\rightarrow W$ be a linear map.

Show that $$\ker \phi =\left \{\sum_{i=1}^n\lambda_ib_i\mid \begin{pmatrix}\lambda_1\\ \vdots \\ \lambda_n\end{pmatrix}\in \textbf{L}(\phi (b_1), \ldots , \phi (b_n))\right \}$$

I have done the following:

Let $v\in V$. Since $(b_1, \ldots , b_n)$ is a basis of $V$, we have that $\displaystyle{v=\sum_{i=1}^n\lambda_ib_i}$.

Then we have that $$v\in \ker \phi \iff \phi (v)=0_W \iff \phi \left (\sum_{i=1}^n\lambda_ib_i\right )=0_W \iff \sum_{i=1}^n\lambda_i\phi (b_i)=0_W$$

Is this correct so far?