ediculeN

2021-10-25

Let $L:V\to W$ be a linear transformation. Explain the meaning of the following statement: The action of the linear transformation L is completely determined by its action on a basis for V.

Talisha

Step 1
Given:
$L:V\to W$ is a linear transformation.
Now, if $\left\{{v}_{1},{v}_{2},\dots ,{v}_{n}\right\}$ is a basis for V such that:
$L\left({v}_{i}\right)={w}_{i}$, for all i,
Then, the action of L on any element of V can be determined.
Step 2
Let v be in V.
Then,
$v={a}_{1}{v}_{1}+{a}_{2}L\left({v}_{2}\right)+\dots +{a}_{n}{v}_{n}$
So,
$L\left(v\right)={a}_{1}L\left(v1\right)+{a}_{2}L\left({v}_{2}\right)+\dots +{a}_{n}L\left({v}_{n}\right)$
$L\left(v\right)={a}_{1}{w}_{1}+{a}_{2}{w}_{2}+\dots .+{a}_{n}{w}_{n}$
Therefore, L(v) is determined.
Thus, “the action of L is completely determined by its action on a basis for V”.

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