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Let $V$ be a vector space and $L(V)$ be the set of all linear transformation $T:V \rightarrow V$. Then we define the addition as $(T_1 +T_2)(v) = T_1(v) + T_2(v) $ and the scalar multiplication as $(cT)(v) =T(cv)$ so that $L(v)$ become a vector space.

Then my question is what is the dimension of $L(v)$ if $dim(V) = n$? I am trying to find the basis of linear transformations but have no idea.

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If $\{v_1 ,.., v_n\}$ is a basis for $V$.

Then, check that, $\{T_{i,j} \}_{1\le i,j \le n}$ is a basis for $L(V)$ where:

$T_{i,j}(v_i) = v_j , T_{i,j} (v_k) = 0$ for $k\ne i$

So, $dim (L(V)) = n^2$.

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