[Math] Extension Theorem and Span

linear algebra

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.

Best Answer

This depends on what exactly you want to know, and what your definition of basis is.

Throughout my answer, I will use a well-known lemma: Let $(v_i)_{i \in I}$ be a family of vectors. Then the following are equivalent:

  1. The $v_i$ form a a linearly independent system of generators.
  2. The $v_i$ form a maximal linear independent system (i.e., adding any vector to the system makes it linearly dependent).
  3. The $v_i$ form a minimal generating system (i.e., removing any vector from it makes it cease to be a generating system). In particular, $v_i \not\in \operatorname{span}(v_1,\ldots,v_{i-1},v_{i+1},\ldots,v_n)$.

Any of these three characterizations can be used to define the basis of a vector space. Usually, one takes the first formulation to define a (Hamel) basis.

The Extension System now states that $(v_1,\ldots,v_k)$ can be extended to a basis $(v_1,\ldots,v_n)$. If you assume that the Extension Theorem holds, your question is simple to answer: since $(v_1,\ldots,v_n)$ is a basis, it is a minimal generating system, by the above characterization.

If you are instead asking about how one proves that the construction of the extension theorem guarantees that $(v_1,\ldots,v_n)$ is a basis, it is again sufficient to point to the lemma: The Extension Theorem describes the existence of a maximal linear indepdent system, and again, this system is a minimal system of generators.

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