# Let T V rightarrow W be a linear transformation Let U be a subspace

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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
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 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 multipl ication. So let two vectors v,w belong to T-1(U ...
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