Let L:V \rightarrow W be a linear transformation. Explain the meaning of t

ediculeN

ediculeN

Answered question

2021-10-25

Let L:VW 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.

Answer & Explanation

Talisha

Talisha

Skilled2021-10-26Added 93 answers

Step 1
Given:
L:VW is a linear transformation.
Now, if {v1,v2,,vn} is a basis for V such that:
L(vi)=wi, 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=a1v1+a2L(v2)++anvn
So,
L(v)=a1L(v1)+a2L(v2)++anL(vn)
L(v)=a1w1+a2w2+.+anwn
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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