# Suppose A and B are n × n matrices. Show that range(AB) ⊆ range(A)

1. Apr 16, 2013

### jazz_hole

1. The problem statement, all variables and given/known data

Suppose A and B are n × n matrices. Show that range(AB) ⊆ range(A)

2. Relevant equations

3. The attempt at a solution

I think I need to show AB is a linear combination of the columns of A, but I'm not sure how to show this in general.

2. Apr 16, 2013

### tiny-tim

welcome to pf!

hi jazz_hole! welcome to pf!
isn't it obvious?

3. Apr 16, 2013

### jazz_hole

I can see how it is true, but I don't know how to show it rigorously. For all n x n matrices.

4. Apr 16, 2013

### vela

Staff Emeritus
Assume $y \in \text{range}(AB)$. Show that $y \in \text{range}(A)$.

You need to understand what is meant precisely when you say $y \in \text{range}(AB)$.

5. Apr 16, 2013

### tiny-tim

start "let y ε range (AB)

then there exists …"

6. Apr 16, 2013

### Staff: Mentor

Moderator note -- The OP has been banned because of his choice of username.