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Let ##V## and ##W## be vector spaces, ##T : V \rightarrow W## a linear transformation and ##B \subset Im(T)## a subspace.

(a) Prove that ##A = T^{-1}(B)## is the only subspace of ##V## such that ##Ker(T) \subseteq A## and ##T(A) = B##

(b) Let ##C \subseteq V## be a subspace. Prove that ##A = Ker(T) \oplus C## iff ##T(C) = B## and ##T|_C## is injective.

The attempt at a solution

Per usual, I'm stuck on the notation here, but I think I have an idea about where the proof comes from, at least in the first part.

To organize my information, I know the following:

##A## is a subspace of ##V## and thus meets all criteria for being a subspace.

##A = T^{-1}(B) | T^{-1} : W \rightarrow V##

As T is invertible, we can deduce that T is bijective as a function and thus both onto and one-to-one, and also that ##V \cong W##. [I have this proof from my notes and previous work]

I also have the definition of kernel and the proof relating it to the transformation's injectivity, also from a previous exercise.

Now, defining the kernel of ##T## :

##Ker(T) = \{v \in V : T(v) = 0\} = T^{-1} (\{0\})##

##T^{-1} (\{0\}) \in T^{-1} \rightarrow Ker(T) \subseteq A##

I can prove that more formally, but does the spirit of the exercise even go in that direction? Similarly, is it using the injectivity from ##Ker(T)## that I prove ##T(A) = B## or can I use the definition of ##T^{-1}## to show that if I apply ##T## to ##T^{-1}(w)## I obtain ##\{w\}## and then use injectivity?

I''ll try to work out the second half of the exercise after the first. What exactly does the notation ##T|_C## mean?

Thanks as always for any and all assistance.

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# [Linear Algebra] Linear transformation proof

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