I have difficulty in understanding the proof of this statement: Let W be a subspace of a finite-dimensional vector space V. Then W is finite dimensional.
The proof goes like this. (Linear algebra, Friedberg, theorem 1.11)
Let dim(V) = n. W contains a nonzero vector v1, and {v1} is a linearly independent subset of W. We continue choosing vectors v2,v3,…,vk, if possible, so that {v1,v2,…,vk} is linearly independent, and adjoining another vector from V results in a linearly dependent set. Since no linearly independent subset of V has more than n vectors, this process stops at k ≤ n. Then {v1,v2,…,vk} is a basis for W, so dim(W) ≤ dim(V ).
The part I cant understand is in the bold text. That part seems seems really vague to me. W is likely to be an infinite set and we may need infinite steps to choose such an independent set! Then it may take forever to choose such set.
I think the bold part needs more explanation. How do you think?
Best Answer
The apparently infinite process must stop with $k$ at most $n$ because by the assumption that $\dim{V} = n$, no subset of $V$ containing more than $n$ vectors is linearly independent.