Question

Let W be an infinite dimensional subspace of the vector space V. Show that V is infinite dimensional.

Answer 1

@helder_edwin

Answer 2

let's suppose that V is finite dimensional, say dim V=n. then any set with more than n vectors would have to be linearly dependent. this would imply that any subspace of V has to have dimension <=n. in particular W should have dimension <=n, i.e., should have finite dimension. this is a contradiction.

Answer 3

Even if W and V both is finite dimensional. W would always have to have a dimension <= to V right?

Answer 4

yes

Answer 5

so then I am confused why they would want us to prove something like this that is just known to be true.

Answer 6

maybe to see if u really grasp the ideas of dimension and of subspace.

Answer 7

im not sure if I will ever fully grasp all of these ideas, but your help makes things much easier to understand that is for sure. thank you again.

Answer 8

u r welcome.