5. Let V, W be finite dimensional vector spaces over K. Let...

5. Let V, W be finite dimensional vector spaces over K. Let $n = \text{dim } V$ and $m = \text{dim } W$. (a) Let $\mathcal{B}, \mathcal{B}'$ be bases of V, W respectively. Prove that the assignment $F \mapsto M_{\mathcal{B}}^{\mathcal{B}'}(F)$ is an isomorphism from $\mathcal{L}(V, W)$ to the vector space of $m \times n$ matrices in K. (b) What is the dimension of $\mathcal{L}(V, W)$ in terms of m and n?

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00:01 Hello everyone in this question v and w are the vector space and t belongs to l of v, w and v1 v2 is the basis for v.

00:08 So the first part of the question is we have to prove t is an isomorphism then t v1 t v2 up to t vn is a basis for w.

00:27 So this is the first part we have to prove.

00:29 So from the given data we will know that dimension of v is n and dimension of w is also n.

00:37 So if we show t v1 t v2 up to t vn is linearly independent in w then we can prove that it is a basis for w.

01:11 So first let us take a1 t v1 a2 t v2 plus an t vn equal to 0.

01:21 So taking e common we will be having a1 v1 a2 v2 plus an vn equal to 0.

01:29 Since they have given t is one to one t v equal to 0 if and only if v equal to 0.

01:39 So this implies a1 v1 a2 v2 plus up to an vn equal to 0.

01:46 So this again implies a1 equal to a2 equal to an equal to 0.

01:53 So since this is 0 we can say that since a1 a2 an are 0 we can say that v1 v2 vn are linearly independent because we have got a1 a2 an 0.

02:11 So this again implies t v1 t v2 up to t vn is linearly independent.

02:21 So since it is linear independent and hence it form a basis for w.

02:32 So the first part is over.

02:33 The second part is vice versa that is we have to assume t v2 v3 up to vn is a basis for w then t is isomorphism.

02:51 So this is the thing we have to do.

02:54 So the proof part will be since t of v1 v2 vn is a basis for w we can say that dimension of w equal to n which is equal to dimension of v.

03:07 So if we show t is one to one then we can prove t is isomorphism since dimension of v and w are same...

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