Let A be an m×nmatrix, and B an m×rmatrix. Then the range of AB is a subspace of the range of

Question

Quizplus · Accepted Answer

The range of AB is the set of all possible linear combinations of the columns of AB.

Let C be a column of AB. Then there exists a column vector x in dimension n such that C = ABx.

Let y be a column vector in dimension r. Then (AB)(y) = AB(y) = A(B(y)).

Since B(y) is a vector in dimension m, it can be written as a linear combination of the columns of B.

That is, there exists a vector z in dimension R^n such that B(y) = Bz.

Therefore, (AB)(y) = A(Bz) = A(B(y)) = AC, where C = Bz.

Thus, every column of AB is a linear combination of columns of A, and therefore, the range of AB is a subspace of the range of A.

Let a Be an M×nmatrix, and B an M×rmatrix