More about Dimension

I thought today I could point out some facts about dimension.

  1. If W is a subspace of a vector space V over a field k, then

\dim_k W\leq\dim_kV.

  1. You now know what it means when someone says we live in “3-dimensional” space. You can find three basis vectors for the entire universe. (Strictly speaking, this is probably false, but it at least looks true in what little piece of the universe I can see. More on that when we talk about manifolds.)
  2. Sometimes you may hear people ask “what is the fourth dimension?” To you, the informed reader, this now appears to be an uninformed question. What they are trying to ask is “what does the 4-dimensional vector space \mathbb R^4 look like, and what is its fourth basis vector?” The first part of the question makes sense, but is hard to answer. The second part doesn’t make sense. Bases are just sets and therefore aren’t ordered. I could make “the fourth basis vector” whichever I want. If you ever get this question, just point somewhere and say “that way.” You can’t be wrong.
Now I’d like to point out that we know enough about vector spaces to deal with Galois theory. That being said, there is a lot more to think about in linear algebra, and I want to do some of that first, so as to let this information sink in.
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