Bases for row and column spaces

**alexmahone** · February 13th 2011, 08:51 PM

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

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

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

**alexmahone** · February 14th 2011, 12:39 AM

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...

**HallsofIvy** · February 14th 2011, 03:51 AM

Originally Posted by alexmahone

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

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

If we write A= BC, then C maps all of $R^3$ into a subspace of $R^2$ 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 $R^3$ into a two dimensional subspace of $R^3$ , it cannot be invertible.

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

Thread: Bases for row and column spaces

LinkBack

Thread Tools

Display

Bases for row and column spaces

Similar Math Help Forum Discussions

Bases over vector spaces.

Question Column-spaces

Row/column space bases

Baes for column, row, and null spaces

vectors and column spaces