The True Statement About Matrix Multiplication: Columns Of Ab As Linear Combinations Of Columns Of B Using Weights From Corresponding Columns Of A.

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.

More Answers:
The Transpose Of A Product Rule For Matrices
Matrix Transpose: Proving A^T + B^T = (A + B)^T.
Mastering Matrix Mathematics: The Distributive Property Of Matrix Multiplication Over Addition.

Error 403 The request cannot be completed because you have exceeded your quota. : quotaExceeded

Share:

Recent Posts

Don't Miss Out! Sign Up Now!

Sign up now to get started for free!