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

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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.
 
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welcome to pf!

hi jazz_hole! welcome to pf! :smile:
jazz_hole said:
Suppose A and B are n × n matrices. Show that range(AB) ⊆ range(A)

isn't it obvious? :wink:
 
I can see how it is true, but I don't know how to show it rigorously. For all n x n matrices.
 
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)##.
 
jazz_hole said:
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 …" :wink:
 
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Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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