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

Kayla Kline

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.

Karen Robbins

c) Given:
$V={M}_{21},W={M}_{32}$
Since, we know that ${M}_{21}$ is a vector space of all $2×1$ matrices and it is 2- dimensional vector space and ${M}_{32}$ is the vector space of all $3×2$ 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.

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