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

[jazz_hole]

Homework Statement

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

Homework Equations

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.

 

[tiny-tim]

welcome to pf!

hi jazz_hole! welcome to pf!

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

isn't it obvious?

 

[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.

 

[vela]

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

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

 

[tiny-tim]

I can see how it is true, but I don't know how to show it rigorously.

start "let y ε range (AB)

then there exists …"

 

[berkeman]

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