Question

For a matrix M, let Rowspace(M) denote the linear span of the rows of M and Colspace(M) denote the linear span of the columns of M. Which of the following hold(s) for all A, B, C ∈ M10(\(\R\)) satisfying A = BC ?

• A. Rowspace(A) ⊆ Rowspace(B)
• B. Rowspace(A) ⊆ Rowspace(C)
• C. Colspace(A) ⊆ Colspace(B)
• D. Colspace(A) ⊆ Colspace(C)

Answer 1

The Correct Option is B, C

- (A) is false because the row space of \( A \) is not necessarily contained in the row space of \( B \) when \( A = BC \). The relationship \( A = BC \) does not impose such a constraint on the row spaces. - (B) is true. Since \( A = BC \), the row space of \( A \) must be contained in the row space of \( C \). This is a property that holds because multiplying by \( B \) does not affect the span of the rows of \( C \). - (C) is true. The column space of \( A \) is contained in the column space of \( B \). This is due to the fact that \( A = BC \) implies that the columns of \( A \) are linear combinations of the columns of \( B \), thus the column space of \( A \) is a subspace of the column space of \( B \). - (D) is false. The column space of \( A \) is not necessarily contained in the column space of \( C \), because the multiplication by \( B \) could affect the linear span of the columns of \( C \). Thus, the correct answers are (B) and (C).