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

Skilled2021-10-26Added 93 answers

Step 1

Given:

$L:V\to W$ is a linear transformation.

Now, if$\{{v}_{1},{v}_{2},\dots ,{v}_{n}\}$ 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”.

Given:

Now, if

Then, the action of L on any element of V can be determined.

Step 2

Let v be in V.

Then,

So,

Therefore, L(v) is determined.

Thus, “the action of L is completely determined by its action on a basis for V”.