# Thread: Bases for row and column spaces

1. ## Bases for row and column spaces

Without multiplying matrices, find bases for the row and column spaces of A:

$\displaystyle A=\left[ {\begin{array}{cc}1 & 2 \\4 & 5 \\2 & 7 \\ \end{array} } \right]\left[ {\begin{array}{ccc}3 & 0 & 3 \\1 & 1 & 2 \\\end{array} } \right]$

How do you know from these shapes that A cannot be invertible?

2. Upon inspection, I find that a basis for the row space of A is (3,0,3) and (1,1,2) while a basis for the column space is (1,4,2) and (2,5,7). Although I'm not sure why...

3. Originally Posted by alexmahone
Without multiplying matrices, find bases for the row and column spaces of A:

$\displaystyle A=\left[ {\begin{array}{cc}1 & 2 \\4 & 5 \\2 & 7 \\ \end{array} } \right]\left[ {\begin{array}{ccc}3 & 0 & 3 \\1 & 1 & 2 \\\end{array} } \right]$
If we write A= BC, then C maps all of $\displaystyle R^3$ into a subspace of $\displaystyle R^2$ which can, of course, have dimension no larger than 2. B then maps that subspace into a subspace of R^3 which can have dimension no larger than 2. Since A= BC maps $\displaystyle R^3$ into a two dimensional subspace of $\displaystyle R^3$, it cannot be invertible.

How do you know from these shapes that A cannot be invertible?