Question

consider the vector space M (2,2), with the standatd operations of matrices addition and scalar multiplication. Let W=(2x2matrix: a,b,c,d where a+d=1. Is W a subspace of M(2,2)

Answer 1

Since \(a+d=1\) we know that not every number in one of the 2x2 matrices in W is 0. Thus, W does not contain the 0 vector. Hence, W is not a subspace.

Answer 2

Thank you. But which axiom should I use?

Answer 3

The axiom that says you have to have the 0 vector.

Answer 4

I found it. Thank you. I just don't know how to show it in this problem.

Answer 5

Ther is another one aI have a problem with. Let A be a 2x2 matrix and W= ( x=(x1,x2): Ax=0)). Show that W is a subspace of the vector space R2.

Answer 6

To show something is a subspace, you need to show 3 things.

1. The potential subspace has the 0 vector.
2. If u and v are elements of W, then any linear combination of u and v is an element of W.
3. If u is an element of W and c is a scalar from \(\mathbb{R}\), then the scalar product cu is an element of W;