1. ## Undergraduate Linear Algebra Question

I'm struggling with where to begin. This is a practice question for my undergraduate Linear Algebra course.

Let f: V -> W be a linear transformation, and let Y be a subspace of W.

a) Prove that X = {v (is an element of) V: f(v) (is an element of) Y} is a subspace of V.

I intuitively feel that this is true - elements of V are being mapped to W, and X contains the elements of V that map to the subspace of Y within W, which suggests that X is within V. However, I have no idea of where to begin with a formal proof.

b) Prove that if f is surjective and V is finite dimensional, then dim V - dim X = dim W - dim Y.

Here I am lost. I understand (or I think I do) surjectivity, and I think that the restriction of f to X gives a linear transformation g: X -> W with g(x) = f(x) for all x (an element of) X, which I'm fairly sure is relevant to the solution, but again, I have no idea with how to go about setting out a formal proof.

I apologise for the "(an element of)", I'm still learning the formatting. Thank you for your time.

2. ## Re: Undergraduate Linear Algebra Question

Originally Posted by Halcyon
I'm struggling with where to begin. This is a practice question for my undergraduate Linear Algebra course.

Let f: V -> W be a linear transformation, and let Y be a subspace of W.

a) Prove that X = {v (is an element of) V: f(v) (is an element of) Y} is a subspace of V.

I intuitively feel that this is true - elements of V are being mapped to W, and X contains the elements of V that map to the subspace of Y within W, which suggests that X is within V. However, I have no idea of where to begin with a formal proof.
Yes, of course, X is within V. That proves that X is a subset of V. To prove that a given subset of a vector space is a subspace, you only have to prove that the subset if "closed under addition" and "closed under scalar multiplication".
That is, if u and v are in X you must prove that u+ v is in X. If u is in X then f(u) is in Y. If v is in X then f(v) is in Y. What can you say about f(u+ v)?
Similarly, if u is in X then f(u) is in Y. What can you say about f(au) for a any scalar?

b) Prove that if f is surjective and V is finite dimensional, then dim V - dim X = dim W - dim Y.

Here I am lost. I understand (or I think I do) surjectivity, and I think that the restriction of f to X gives a linear transformation g: X -> W with g(x) = f(x) for all x (an element of) X, which I'm fairly sure is relevant to the solution, but again, I have no idea with how to go about setting out a formal proof.

I apologise for the "(an element of)", I'm still learning the formatting. Thank you for your time.[/QUOTE]