You should show the work that you've attempted so we know where you're stuck. I'll do the first one:
If and , then
If and is a scalar, then
So it is a subspace.
Determine whether the following three sets are a subspace of Rn. If it is, show that the the 3 subspace properties are satisfied, and if it is not, show by example that one of the properties fails.
1. The set of solutions of Ax= x, where A is any n x n matrix.
2. The first quadrant in R2. That is: {[x_1, x_2] ∈ R2| x_1 ≥ 0 and x_2 ≥ 0}
3. {[x_1, x_2] ∈ R2| (x_1 ≥ 0 and x_2 ≥ 0) or (x_1 0 ≤ and x_2 ≤ 0)}