Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A.
False because the roles of A and B should be reversed in the second half of the statement.
The statement Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A is true.
When we multiply a matrix A by a matrix B, the resulting product AB has columns that are linear combinations of the columns of B using weights from the corresponding columns of A. This is a consequence of the way matrix multiplication is defined.
To see why this is true, consider the jth column of AB. This column is obtained by multiplying the matrix A by the jth column of B, which is denoted as Bj. In other words, the jth column of AB is given by:
AB[:,j] = A * Bj
Expanding this expression using the definition of matrix multiplication, we get:
AB[:,j] = [A[:,1] A[:,2] … A[:,n]] * [b1[j] b2[j] … bn[j]]^T
= A[:,1]*b1[j] + A[:,2]*b2[j] + … + A[:,n]*bn[j]
This shows that the jth column of AB is a linear combination of the columns of B using weights from the corresponding column of A. Each term in the sum corresponds to multiplying the jth column of B by a scalar from the jth column of A. Therefore, the statement is true.
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