Subspaces Recall that a nonempty subset W of a vector space

Subspaces: Recall that a nonempty subset W of a vector space V is a subspace of V if and only is the following two conditions are satisfied for any vectors u, v in W and any scalars c in R:

(I) If u and v are vectors in W, then the vector u+v is also in W.

(II) If u is a vector in W and c is a scalar, then the vector cu is also in W.

Are the following set of vectors W a subspace of V? Give either a proof (if you think it is true) or a counterexample (if you think it is false). Counter examples without explanation/justification will receive zero credit.

(c) V=R⁴; W is the set of vectors (a, b, y) in R³ such that 100a-37b+pi=0.

(d) V=R³; W is the set of vectors (a, b, y) in R³ such that a-b=-y, a+2b=y.

Solution

(c) Let us suppose that W is a vector space. Let (a,b,y) and (c,d,x) W. Then, we have 100a - 37b+ = 0 and 100c - 37d + = 0. Also (a,b,y) + (c,d,x) = ( a+ c, b +d, y + x) as both (a,b,y) and (c,d,x) W which is a vector space . We have 100(a + c) - 37( b+d ) + = (100a - 37b + ) + (100c- 37d ) = 0 +   (100c- 37d ) =(100c- 37d ) =

(100c- 37d + ) - = 0 - = - . Thus W is not closed under addition. Therefore, W is not a vector space.( If, however, we write y in place of , W will be closed under addirtion. We can also, then easily show that W is closed under scalar multiplication. ( Even if we let 100a - 37b+ = 0 , then also W is closed under scalar multiplication)

(d) Let us suppose that W is a vector space. For all (a,b,y) W, we have a - b = - y and a + 2b = 2y . Let (a,b,y) and (c,d,x) W. Then, we have a - b = - y and a + 2b = 2y . Also, c - d = - x and c + 2d = 2x . Now,  (a,b,y) +  (c,d,x) = ( a+ c, b +d, y + x) as both (a,b,y) and (c,d,x) W which is a vector space . Then (a +c) - (b +d) = (a -b) + (c - d) = -y -x = - (y +x) . Also (a + c) + 2(b +d) = (a + 2b) + (c +2d) = 2y + 2x = 2 (y +x). Thus W is closed under addition. Now let p be a scalar. We have p(a , b, y) = (pa,pb, py) as W is a vector space.Again pa -pb = p(a -b) = P(-y) = -py . Also, pa +2pb = p (a +2b) = p(2y) = 2(py). Thus W is also closed under scalar multiplication. Thus W is a vector space.