Kathy Williams

2022-01-05

Find the dimension of the vector space U of all linear transformations of V into W for each of the following:

(a)$V={R}^{2},W={R}^{3}$

(b)$V={P}_{2},W={P}_{1}$

(c)$V={M}_{21},W={M}_{32}$

(d)$V={R}_{3},W={R}_{4}$

(a)

(b)

(c)

(d)

Kayla Kline

Beginner2022-01-06Added 37 answers

a)Given:

$V={R}^{2},W={R}^{3}$

Since$R}^{2$ is 2-dimensional vector space and $R}^{3$ is 3-dimensional vector space, thus, $U$ is isomorphic to $M}_{32$ .

Hence, the dimension of$U$ is 6.

b) Given:

$V={P}_{2},W={P}_{1}$

Since we know that$P}_{2$ is the vector space of polynomial of degree 2.

Also, it is 3-dimensional vector space

And$P}_{1$ polynomial of degree 1 and it is 2-dimensional vector space, thus, $U$ is isomorphic to $M}_{23$

Hence, the dimension of$U$ is 6.

Since

Hence, the dimension of

b) Given:

Since we know that

Also, it is 3-dimensional vector space

And

Hence, the dimension of

Karen Robbins

Beginner2022-01-07Added 49 answers

c) Given:

$V={M}_{21},W={M}_{32}$

Since, we know that$M}_{21$ is a vector space of all $2\times 1$ matrices and it is 2- dimensional vector space and $M}_{32$ is the vector space of all $3\times 2$ matrices and it is 6- dimensional vector space.

Hence, the dimension of$U$ is 12.

d) Given:

$V={R}_{3},W={R}_{4}$

Since,$R}_{3$ is a 3- dimensional vector space and $R}_{4$ is a 4- dimensional vector space.

Thus,$U$ is isomorphic to $M}_{43$ .

Hence, the dimension of$U$ is 12.

Since, we know that

Hence, the dimension of

d) Given:

Since,

Thus,

Hence, the dimension of

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