Determine whether the set x1 x2x1 x2 0 forms a subspace of

Determine whether the set {[x_1 x_2]|x_1 x_2 = 0} forms a subspace of R^2. Determine whether the set of all 2 times 2 diagonal matrices forms a subspace of M_2, 2 (the vector space consisting of all 2 times 2 matrices with the usual addition and scalar multiplication).

Solution

a)

No

Consider two elements in thsi set

[0 1]^T and [1 0]^T

BOth belong to this set

BUt their sum which is [1 1]^T does not

Hence it is not a subspace

b)

Yes.

1. 0 belongs to this set

2. Sum of any 2 diagonal matrices is a diagonal matrix hence closed under addition.

Hence it forms a subspace