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Given $T : V \rightarrow W$ as in Exercise $35,$ and given a subspace $Z$ of $W,$ let $U$ be the set of all $\mathbf{x}$ in $V$ such that $T(\mathbf{x})$ is in $Z$ Show that $U$ is a subspace of $V .$

$U$ is a subspace of $V$

Calculus 3

Chapter 4

Vector Spaces

Section 2

Null Spaces, Column Spaces, and Linear Transformations

Vectors

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in this problem there's a linear transformation. Greater acting crawlspace be onto space. Definitely. And in space stuff we have subspace, etc. And we're gonna say let you represent all Xing We anywhere trying to prove that you is a subspace No, We're going to say that you won And you two are two elements in space, you and tur. And to you too we know And this problem settling This is given that these Aaron Z since t easily near a praetor concert t you on wants to see you too is t you want waas You too So this your wants you to is all saying z now You want any too, aren't you? So from this we know that you want waas you jer should be in you So this means that you is blows on there back tradition No, we're gonna use you want to you So we know that do you want isn't easy So if you multiply that with any constant and this should be all saying z right And since I can't easily you to information this will be the tee off. See you are And it should be also Z No, you want wasn't see? Sorry. You always you. It means that see, you won should be in you. And from this we see that here is close under back to for protection. So these are the requirements that we need for subspace subspace. So using this, we can say that you is a subspace. Oh, be

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