Recent content by harvesl

Thanks!

Ah yeah, that makes more sense - define W' and then state it clearly is a subspace. Also, for the first question regarding showing there is a subspace of dimension r for each r between 0 and n - can this also be best done by extending the basis n times from the 0-dimensional subspace S = Sp{0}...

Yeah, I've redone the second proof - currently working on the first. U = Sp\left\{u_1, u_2, ... , u_k\right\} Where \left\{u_1, u_2, ... , u_k\right\} is linearly independent We can extend this to a basis of W as U \subseteq W So for w_{k+1}, w_{k+2}, ... , w_l \in W \backslash U...

Hi, I'd be grateful if someone could tell me whether these proofs I've done are correct or not. Thanks in advanced. Let V be an n-dimensional vector space over \mathbb{R} Prove that V contains a subspace of dimension r for each r such that 0 \leq r \leq n Since V is n-dimensional...

Thanks.

Homework Statement Assume V = \mathbb{R}^n where n \geq 3. Suppose that U,W,X are three distinct subspaces of dimension n-1; is it true then that dim(U \cap W \cap X) = n-3? Either give a proof, or find a counterexample.The Attempt at a Solution The question previous to this was showing that...

Thanks, I've solved this now. One good counter example is to Let M = \left\{(x,0) | x \in \mathbb{R}\right\} Let N = \left\{(0,y) | y \in \mathbb{R}\right\} and let L be any line through the origin. Which gives you that L \cap (M + N) is the set of all points on the line L...

Homework Statement Let L,M,N be subspaces of a vector space V Prove that (L \cap M) + (L \cap N) \subseteq L \cap (M + N) Give an example of subspaces L,M,N of \mathbb{R}^2 where (L \cap M) + (L \cap N) \neq L \cap (M + N) Homework Equations The Attempt at a Solution...

Homework Statement Let A be an n x n matrix where n \geq 2. Show that A^{\alpha} = 0 (where A^{\alpha} is the cofactor matrix and 0 here denotes the zero matrix, whose entries are the number 0) if and only if rankA \leq n-2 Homework Equations The Attempt at a Solution No idea...