Question

**Let A be an invertible 'n x n' matrix and B be a 'n x m' matrix.**
Find A and B so that the column space of AB and the column space of B are different.
((so stuck~~))

Answer 1

First of all product AB is defined or not;
Let's see: -
\[n \times \color{red}{\cancel n} = \color{red}{\cancel n} \times m.\]yes it is defined.
but of what order would AB be: ?
Let's see: - \[[n] \times \color{red}{\cancel n} = \color{red}{\cancel n} \times [m].\]It is \[[n \times m]\]As u can see.

Answer 2

So it is a question based on "rank of the matrix".
I didn't noticed it before.
Actually i also haven't studied the rank stuff yet
so really sorry!
:)

Answer 3

May be it is of some help:)



http://en.wikipedia.org/wiki/Column_space