How Can Proofs Enhance Understanding of Linear Algebra Theorems?

[elle]

Hi,
I'm currently self-studying Linear Algebra and I've come across a few Theorems in the text that I am reading. However, a lot of them just gives the general theorem and statements without proofs and personally I find looking through derived proofs give me a better understanding of the Theorem itself

I would very much appreciate if someone can help show the proofs of the following Theorems:

http://i26.photobucket.com/albums/c109/mathsnerd/txt.jpg"

http://i26.photobucket.com/albums/c109/mathsnerd/txt2.jpg"

Thanks very much!

 

[Office_Shredder]

For question 1, what does S have to do with the question? Am I just confused?

To start, you should have learned a theorem that a minimal spanning set is a maximal linear independent set is a basis of V. So if U is a subspace of V, then the basis of V spans U, so the number of elements in a basis of U must be less than or equal to the elements in the basis of V, otherwise the above theorem is wrong. Then the definition of dimension kicks in, and you're good.

For the part if dimU=dimV, U=V, I would start with the Steinitz exchange procedure. If you don't know what it is, look it up, it's a neat theorem

For the second, definitely start with the additivity property of inner products... a little bit of algebra should sort you right out