Question

part a and b

PROBLEM (HAND-IN ASSIGNMENT) Use the Subspace Test to determine whether the following sets W are subspaces of the given vector spaces: (A) The set W to be of all triples of real numbers (x, y, z) satisfying that 2x - 3y + 5z = 0 with the standard operations on Ris a subspace of R3. (B) The set of all 2 x 2 invertible matrices with the standard matrix addition and scalar multiplication.

Answer 1

(A). If 2x-3y+5z = 0, then 2x = 3y-5z so that x = (3y-5z)/2.

Thus, every vector in W is of the form ((3y-5z)/2,y,z).

Let X = ((3a-5b)/2,a,b) and Y = ((3c-5d)/2,c,d) be 2 arbitrary vectors in W and let k be an arbitrary real scalar.

Then, X+Y = ((3a-5b)/2,a,b)+((3c-5d)/2,c,d)= ( (3(a+c)-5(b+d))/2, (a+c), (b+d)). This implies that X+Y ∈W so that W is closed under vector addition.

Also, kX = k((3a-5b)/2,a,b)= ((3ka-5kb)/2,ka,kb). This implies that kX ∈W so that W is closed under scalar multiplication.

Further, when a = b = 0, then X = 0. This implies that the zero vector (0,0,0) ∈W.

Hence W is a vector space and, therefore, a subspace of R³.

(B). Let the given set be denoted by W.

Let A =

1	0
0	1

and B =

-1	0
0	-1

Then det(A) = det(B) = 1 ≠ 0 so that both A and B are invertible. Hence, A, B ∈W.

However, A+B =

0	0
0	0

so that det(A+B) = 0. Therefore, A+B is not invertible. Hence, A+B ∉ W so that W is not closed under vector addition.

Therefore W is not a vector space and, therefore, not a subspace of M_2x2 , the vector space of all 2 x 2 matrices.