Linear algebra MOC

Vector subspace

A vector subspace 𝑆 ≤𝑉 of a vector space 𝑉 is a subset 𝑆 ⊆𝑉 that is a vector space under the same scalar multiplication and vector addition. linalg This can be boiled down to the following requirement:

If ⃗𝐮,⃗𝐯 ∈𝑆 and 𝜆,𝜇 ∈𝕂, then 𝜆⃗𝐮 +𝜇⃗𝐯 ∈𝑆.

The concept of subspaces naturally leads to the concept of a Span, which is the smallest possible subspace containing a set of specific vectors within the main vector space.

Properties

1. The subspaces of a given vector space form a Complete lattice with initial {⃗𝟎} and terminal 𝑉. The greatest lower bound is the intersection of subspaces, the least upper bound is the sum of subspaces.
2. A nontrivial vector space 𝑉 over an infinite field 𝕂 is not the union of finitely many proper subspaces.¹

Proof of 2

Let 𝑉 be a nontrivial vector space over 𝕂. Assume 𝑉 =⋃𝑛𝑖=1𝑆𝑖 and without loss of generality 𝑆1 ⊈⋃𝑛𝑖=2𝑆𝑖. Now let 𝑣 ∉⋃𝑛𝑖=2𝑆𝑖 and 𝑤 ∉𝑆1. Then the infinite set

𝐴={𝜆𝑤+𝑣:𝜆∈𝕂}

is an infinite set corresponding to the line through 𝑣 parallel to 𝑤. We will show that 𝐴 contains at most one element from each 𝑆𝑖 and must thence be finite, leading to contradiction.

First note that if 𝜆𝑤 +𝑣 ∈𝑆1 for 𝜆 ≠0 then 𝑣 ∈𝑆1 since 𝑤 ∈𝑆1, contradicting our assumption. Next, suppose for some 𝜆1 ≠𝜆2 we have 𝜆1𝑤 +𝑣 ∈𝑆𝑖 and 𝜆2𝑤 +𝑣 ∈𝑆𝑖, Then

(𝜆1−𝜆2)𝑤=(𝜆1𝑤+𝑣)−(𝜆2𝑤+𝑣)∈𝑆𝑖

so 𝑤 ∈𝑆𝑖, which is also a contradiction.

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Footnotes

1. 2008. Advanced Linear Algebra, p. 39 ↩