this algebra proof make sense?

Question

let V be a finite dimensional vector space, M and N linear subspaces of V, prove

dimM+dimN = dim(M+N) + dim(M n N)

n = intersection,


proof:
First prove the lemma
If U is a subspace of V and {u_1,..., u_k} is a base in U then there are some vectors v_1,..., v_l such that {u_1,..., u_k, v_1,..., v_l} is a base in V.

Now choose the base B in M n N. Extend it by lemma to a base Bm in M, and base Bn in N. Now prove that Bm u Bn is a base of M+N.

Let #X be a number of elements in X.
You have
dim M = #Bm
dim N = #Bn
dim (M+N) = #(Bm u Bn)
dim (M n N) = #B
and #(Bm u Bn) = #Bm + #Bn - #(Bm n Bn) = #Bm + #Bn - #B because Bm n Bn = B



does that look right, any help is appreciated

Answer 1

Yes, that looks exactly right. Now just expand the sketch into a full proof, and you'll be done.

Answer 2

that's fake counterexamples are accessible to discover working occasion, take V=R^2 with ordinary initiating place e1=(0,a million), e2=(a million,0) Any line that passes by potential of skill of using 0 regardless of the certainty that's no longer an axis is a vectorial area. of direction those do now no longer look generated by potential of skill of way of e1 or e2 In usually happening, evaluate V had initiating place u,v The subspace generated by potential of skill of way of u+v has length a million regardless of the certainty that's no longer generated by potential of skill of way of u or v edit: i used to be typing on an same time you revealed. supply up commenting. i do now no longer care approximately components and BA as much as you

Answer 3

Looks good.