Question

If A is an n×n matrix and b≠0 in Rn, then consider the set of solutions to Ax=b. Select true or false for each statement. 1. This set is a subspace 2. This set is closed under scalar multiplications 3. The set contains the zero vector 4. This set is closed under vector addition

Answer 1

Check the explanation

Explanation:

1. FALSE. If X and Y are 2 solutions to the equation AX = b, then A(X+Y) = AX +AY = b+b = 2b ≠ b as b ≠ 0. This implies that X+Y is not a solution to the equation AX = b. Hence the set of solutions to the equation AX = b is not closed under vector addition.

2. FALSE. If X is a solution to the equation AX = b and if k is an arbitrary scalar other than 1, then A(kX) = kAX = kb≠ b. This implies that kX is not a solution to the equation AX = b. Hence the set of solutions to the equation AX = b is not closed under scalar multiplication.

3. FALSE, unless b = 0. A.0 = 0 . Since b ≠ 0, the set of solutions to the equation AX = b does not contain the 0 vector.

4. FALSE. In view of 1,2,3 above, the set of solutions to the equation AX = b is not a subspace since b ≠ 0.