# span, basis and dimension

• Mar 20th 2011, 06:44 AM
wik_chick88
span, basis and dimension
im having a little/big problem with the whole concept of span, basis and dimension
eg. $\displaystyle S = \{\left(\begin{array}{cc}a&b\\c&a\end{array}\right ) \in V: a,b,c \in \Re \}$
how would i go about finding the span, the basis and the dimension of $\displaystyle S$?
• Mar 20th 2011, 07:54 AM
Plato
Quote:

Originally Posted by wik_chick88
im having a little/big problem with the whole concept of span, basis and dimension
eg. $\displaystyle S = \{\left(\begin{array}{cc}a&b\\c&a\end{array}\right ) \in V: a,b,c \in \Re \}$
how would i go about finding the span, the basis and the dimension of $\displaystyle S$?

Is this a basis: $\displaystyle \[ \left\{ {\left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right),\,\left( {\begin{array}{*{20}c} 0 & 1 \\ 0 & 0 \\ \end{array} } \right),\,\left( {\begin{array}{*{20}c} 0 & 0 \\ 1 & 0 \\ \end{array} } \right)} \right\}~?$
• Mar 20th 2011, 09:25 AM
HallsofIvy
$\displaystyle \begin{pmatrix}a & b \\ c & a\end{bmatrix}= \begin{pmatrix}a & 0 \\ 0 & a\end{pmatrix}+ \begin{pmatrix}0 & b \\ 0 & 0\end{pmatrix}+ \begin{pmatrix}0 & 0 \\ c & 0\end{pmatrix}$

It makes no sense to talk about finding the "span" of a vector space.