Determine if U={(x,y)∈ R^2:y=-5x=5 is a subspace of R^2

Question

Determine if U={(x,y)∈ R^2:y=-5x=5 is a subspace of R^2

Answer 1

I assume that you mean y = -5x + 5

Let (a,b) and (c,d) be elements of U. Is (a,b) + (c,d) also in U? If so, then b+d = -5(a+c) + 5

Since (a,b) is in U, b = -5a+5 and since (c,d) is in U, d = -5c+5

(a,b) + (c,d) = (a+c, b+d)

b+d = (-5a+5) + (-5c+5) = -5(a+c) + 10 ≠ -5(a+c) + 5. Thus, (a,b) + (c,d) is not in U.

So, U is not a subspace of lR².