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Suppose $V$ and $W$ are finite-dimensional. Let $v \in V$. Let $E = \{T \in \mathsf{L}(V, W) \mid Tv = 0\}$.

(a) Show that $E$ is a subspace of $\mathsf{L}(V, W)$.

(b) Suppose $v \ne 0$. What is $\dim E$?

Part a) is not a problem for me. Part b) on the other hand... I was given a hint that if $v$ is not equal to $0$ then we can extend it to a basis for $V$. We can also fix a basis for $W$. I just don't know how to use this hint to solve part b? Any help would be appreciated!

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    Do you know how to find the matrix of a linear transformation with respect to a certain basis? – 2017-02-09
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    Do you know what the dimension of $Hom(U,V)$ is? – 2017-02-09
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    @Omnomnomnom yes I do! – 2017-02-09

1 Answers 1

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Hint: If $\{v_1,\dots,v_n\}$ is a basis and $\{w_1,\dots,w_m\}$ a basis for $W$, then the transformations $f_{ij}$ satisfying $$ f_{ij}(v_k) = \begin{cases} w_j & k=i\\ 0 & k \neq 0 \end{cases} $$ form a basis of $L(V,W)$.

Alternatively: if we fix the above bases with $v_1 = v$, then $L(V,W)$ is identified space of $m \times n$ matrices (of transformations with respect to this basis). We can identify $E$ with those matrices whose first column is zero.

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    I don't understand how to write out that transformation. That's my problem – 2017-02-09
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    See my latest edit. – 2017-02-09