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Linear Algebra and Its Applications | 5th Edition | ISBN: 9780321982384 | Authors: David C. Lay; Steven R. Lay; Judi J. McDonald
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Given as in Exercise 35, and given a subspace Z of W, let

Chapter 4, Problem 36E

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QUESTION:

Problem 36E

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

Reference:

Let V and W be vector spaces, and let  be a linear transformation. Given a subspace U of V, let T (U) denote the set of all images of the form T (x), where x is in U. Show that T (U) is a subspace of W.

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QUESTION:

Problem 36E

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

Reference:

Let V and W be vector spaces, and let  be a linear transformation. Given a subspace U of V, let T (U) denote the set of all images of the form T (x), where x is in U. Show that T (U) is a subspace of W.

ANSWER:

Solution 36E

Step 1 of 2

Supposeis a linear transformation, where are vector spaces.

Let Z be the subspace of W.

The set  is defined as.

The objective is to prove that is a subspace of V.

Since T is a linear transformation and, so the zero vectorof W is in.

Let

Then

Since Z is a subspace of W.

So Z is closed under vector addition.

Thus,

Now Since T is a linear transformation

Therefore,

Thus,

Therefore, U is closed under vector addition.

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