Question

Let A be an m x n matrix. For which columns b

in R^m is U = {x | x in R^n, Ax = b) a subspace of
R^n? Support your answer.

Answer 1

The set U = x in R^n, Ax = b is a subspace of R^n if and only if b is the zero vector in R^m, because a subspace must contain the zero vector, and it must be closed under addition and scalar multiplication, which is not the case for any non-zero b.

Step-by-step explanation:

The question is related to the field of linear algebra, specifically about linear transformations and subspace criteria. Let A be an m x n matrix. The set U = x in Rn, Ax = b) will be a subspace of Rn only if b is the zero vector in Rm. This is because one of the properties of a subspace is that it must contain the zero vector, meaning Ax = 0 must have a solution within the set.

Additionally, for a non-zero vector b, the solutions to Ax = b do not necessarily include the zero vector, and thus do not form a subspace.

Furthermore, for any set to be considered a subspace, it must also be closed under addition and scalar multiplication. The set U defined by Ax = b for a non-zero b is not closed under addition or scalar multiplication, because if you take any two vectors x1 and x2 such that Ax1 = b and Ax2 = b, then A(x1 + x2) will not equal b, but rather 2b, violating the subspace criterion of closure under addition.

Similarly, for any scalar c, A(cx) will equal cb and not b, violating the subspace criterion of closure under scalar multiplication.