span, basis and dimension

**wik_chick88** · March 20th 2011, 06:44 AM

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

**Plato** · March 20th 2011, 07:54 AM

Originally Posted by wik_chick88

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

Is this a basis: $\[ \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\}~?$

**HallsofIvy** · March 20th 2011, 09:25 AM

$\begin{pmatrix}a & b \\ c & a\end{bmatrix}= \begin{pmatrix}a & 0 \\ 0 & a\end{pmatrix}+ \begin{pmatrix}0 & b \\ 0 & 0\end{pmatrix}+ \begin{pmatrix}0 & 0 \\ c & 0\end{pmatrix}$

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

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