Let T: V rightarrow W be a linear transformation. Let U be a subspace of W. Show that its pro-image T-1(U) = {v V | T(v) U} is a subspace of V.
Solution
In this question, we just need to show that T -1 (U) is closed under addition, and scalar multiplication.
So let two vectors v,w belong to T -1 (U); this means that Tv as well as Tw belong to U (by def). Since U is a subspace of W, the sum - Tv+Tw again belongs to U. But since T is linear, we can write T(v+w) belongs to U. And hence v+w belongs to T -1 (U).
Similarly it is very clear that if v belongs to T -1 (U), and \'c\' is a scalar, then cv also belongs T -1 (U).
I hope this helps. Do reply, if you need more explanation.
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Let T- V rightarrow W be a linear transformation- Let U be a subspace.docx
1. Let T: V rightarrow W be a linear transformation. Let U be a subspace of W. Show that its pro-
image T-1(U) = {v V | T(v) U} is a subspace of V.
Solution
In this question, we just need to show that T -1
(U) is closed under addition, and scalar
multiplication.
So let two vectors v,w belong to T -1
(U); this means that Tv as well as Tw belong to U (by def).
Since U is a subspace of W, the sum - Tv+Tw again belongs to U. But since T is linear, we can
write T(v+w) belongs to U. And hence v+w belongs to T -1
(U).
Similarly it is very clear that if v belongs to T -1
(U), and 'c' is a scalar, then cv also belongs T -1
(U).
I hope this helps. Do reply, if you need more explanation.
