Anyone at all :\?
Let A be a real invertible x matrix. Show that
<x,y> ≡ =
defines an inner product in ℝ^n, where x and y are column vectors in ℝ^n. What happens when A is not invertible? (Note : is the transpose of matrix M, obtained by interchaging the rows and colums of M).
Let .
First one :But is a real numbers (i.e. matrix) hence and we get what we want.
Last one :
Denote , and .
- If , we check that because
- If hence, for all such that , . This exactly means that is in the nullity of . As is invertible, .
There may have a problem of notation : I write the transpose of . Anyway :
(that's the definition)
because for any two matrices and .
hence because (previous rule with and )
No. These ones are really basic questions that you should be able to answer by yourself. The only things you need to know is that and that with and . Try to do it and, if you want, post your answer, I'll tell you if it's correct or not.Can you show me how to do the other two as well ? Properties 2 & 3.
Okay, I have a slightly clearer view of it now considering the confusion was about the notation. Hm, to prove Property 1 only require that two steps ?
I don't know how to explain how did it go from <x,y> to <y,x>.
I am quite doomed right now, this is my assignment question which dues tomorrow and I am pretty much still quite blur about it
We got and we also know that is a real number so it can be seen as the one-one matrix . This gives us .
Hence .
Does your assignment only depend on one question ?I am quite doomed right now, this is my assignment question which dues tomorrow and I am pretty much still quite blur about it