Question

I need help understanding subspaces in linear algebra. Given the vector space R^2X2 (where R is all real numbers) I would think that this would include ALL 2X2 matrices with real numbers- but this isnt the case. Why? For example, my book says that the set of all 2X2 triangular matrices, matrices in A where a12 = 1, and singular matrices are not a subspace of R^2X2. Why is this? Thank you.

Answer 1

The Vector Space R^(2x2) is the set of all 2x2 matrices - with real numbers as you said, however in order for a subset 'T' to be a subspace, four conditions must be satisfied, namely;
1) your subset T must be non-empty
2) for any v,w E T, v+w E T
3) An identity must exist
4) Scalar multiplication

So if your book states that triangular and singular matrices are not subspaces; then they mustn't comply with one or more of the conditions

I would recommend maybe taking an example of both a triangular and singular matrix and trying it for yourself; it's the best way to understand it if you can visually work through an example

I hope this helps a bit! :)

Answer 2

thank you! that makes perfect sense with the rules you listed...my book never mentioned that an identity must exist. it just listed the general "axioms" to follow regarding associative laws, additive inverses, closure properties, etc. Does the condition "identity must exist" stem from some previous reasoning? thanks again for the help!

Answer 3

It comes from the conditions satisfying a Vector Space, as a subspace is in fact a vector space in itself; albeit a somewhat reduced one

So in order for a Set 'V' to be a Vector Space; it must satisfy a few conditions (not too dissimilar from the subspace ones) - however the identity element is crucial

So there must exist an element 'e' such that if you multiply any element of your set by 'e' it will still belong in your set; and also intertwines itself with the inverse

As an element 'x' * 'x(^-1)' = e

The best way I find to picture it is to create some 2x2 matrices and practice finding inverses; but note that the identity of a 2x2 matrix in the set R^2x2 is always the same ie:
(1,0)
(0,1) I hope that showed in the appropriate order!

Answer 4

ahh okay got it! thank you for the response!