Question

If the columns of an m x n matrix A span Rᵐ, then the equation Ax=b is consistent for each b in Rᵐ. Choose the correct answer below.

A. True. The equation is always consistent in this case.
B. False. The equation is inconsistent for some b in Rᵐ.
C. True. The equation is inconsistent for any b in Rᵐ.
D. False. The equation is never consistent in this case.

Answer 1

Option (A), The equation Ax=b is always consistent when the columns of an m x n matrix A span Rᵐ.

Step-by-step explanation:

Answer: A. True. The equation is always consistent in this case.

To understand why, let's first define some terms. The columns of an m x n matrix A span Rᵐ means that the vectors formed by the columns of matrix A can create any vector in Rᵐ through linear combinations. When we have a consistent equation Ax=b, it means that there exists a solution for every vector b in Rᵐ.

In this case, since the columns of matrix A span Rᵐ, every vector in Rᵐ can be expressed as a linear combination of the columns of A. Therefore, the equation Ax=b will always have a solution for every vector b in Rᵐ, making the answer True.