Show that S is a subspace of M2,2 or that it is not given x=w ???

Question

please help i dont understand, that is all the information that is given, the answer says it is a subspace but i cant see why please anyone who knows linear algebra .. thanks

Answer 1

M2,2 is the space of all 2 by 2 real matrices. I'm guessing that the question is whether the set of 2 by 2 matrices [x, y; z, w] with x = w constitute a subspace? Since this is clearly a subset of M2,2, you just need to show (a) that the identity is included and (b) the set is closed under the operation, here matrix addition. These are easily seen to be true: I = [1,0;0,1] satisfies x = w, and [x,y;z,x] + [p,q;r,p] = [x+p, y+q; z+r, x+p ], that is, the sum of two matrices with equal diagonal terms has equal diagonal terms.

Answer 2

I need a little more information in order to help.

What is M2,2 ? What is the space? And what are x and w? elements of what?

The concept of spaces is very broad: lots of things fit the concept: vector space, spaces of matrices, functions, all kinds of things. Elaborate on you question.