Intuition/Picture - Theorems on Linear Independence, Span, Basis, Dimension [Poole, Section 6.2]

Question

I'd like to ask about the intuitions for these theorems, absent in David Poole's Linear Algebra (to which the page numbers refer). Also, are there pictures for these theorems?

Answer 1

I don't think these are easily visualized, but perhaps I can offer some help with intuitions.

Theorem 6.7 says that linear independence is an intrinsic (geometric?) property of a set of vectors. It doesn't matter how you represent them (that is, which basis you use to represent them).

I don't think Theorem 6.9 is properly called the Steinitz exchange lemma. The Steinitz exchange lemma I am familiar with is different. The theorem you have quoted says that the dimension of a vector space is invariant. It is an intrinsic property of the vector space. No matter how you choose a basis, it will always have the same number of elements.

Perhaps the best way to understand 6.10 is just working through the implications between each part of it. The general idea is that if a vector space has dimension $n$, then $n$ is the 'sweet spot'. If you have more than $n$ vectors, you have too many: some will be redundant, so they will be linearly independent. But if you have less than $n$, you have too few: they will not span the space. But if everything is just right (you have $n$ vectors and they are all linearly independent), you get a basis.

6.11 makes two claims. Part (a) says that if $V\subset W$, then $V$ has dimension at most $\dim W$. This should be intuitive. How could a high dimensional subspace be a subset of a low dimensional one? Think of dimension as a kind of size. Part (b) says that if $V$ is an $n$ dimensional space, then the only $n$ dimensional subspace of $V$ is $V$ itself.