# Bases for row and column spaces

• Feb 13th 2011, 08:51 PM
alexmahone
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?
• Feb 14th 2011, 12:39 AM
alexmahone
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...
• Feb 14th 2011, 03:51 AM
HallsofIvy
Quote:

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.

Quote:

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