1
$\begingroup$

By Extension Theorem: you can extend list of linearly independent vectors $(v_1,v_2,\ldots,v_k)$ to a basis $(v_1,v_2, \ldots, v_k, v_{k+1}, v_n)$

How to prove: $v_i$ for $i = 1,2,...,k$ $\notin span(v_{k+1},...,v_n)$?

Thanks.

Edit: Okay, perhaps I should state the full question.

Suppose that W is a subspace of a finite-dimensional vector space V.

(a) Prove that there exists a subspace W and a function T: V → V such that T is a projection on W along W'.

I want to let W: $(v1, \ldots , vk)$ set of linearly independent vectors. W' = $span(vk+1, \ldots, vn)$

So that $V = W \oplus W'$ (direct sum)

Does this work? The definition of basis that I am using is a list of vectors $(u1,\ldots,un)$ that is linearly independent and spans V.

3 Answers 3