Linear Algebra subspace Given that the set M2R of all real 2

Linear Algebra subspace:

Given that the set M_2(R) of all real 2 by 2 matrices is a vector space, (with usual operations), determine whether or not the set of all matrices of the form (a b c -a) is a subspace. If it is a subspace, give its dimension. If it is not a subspace, explain why not.

Solution

The set of matrices form a vector spacesince it is closed under addition and scalar multiplication

On addition

[a1 b1; c1 -a1] + [a2 b2; c2 -a2] = [(a1+a2) (b1+b2); (c1+c2) -(a1+a2)]

On Scalar multiplication

k[a1 b1; c1 -a1] = [ka1 kb1; kc1 -ka1] = [a2 b2; c2 -a2]

Since the vector space is closed under addiiton and scalar multiplication, hence it is a subspace