"Show that
[1]
[1]
[1] can't be written as a linear combination of
[1]
[0]
[1]
and
[0]
[1]
[1] "
I'm not sure where to even start here. Sorry about the formatting I don't know how to make matrices.
Suppose that $\alpha \left[ {\begin{array}{*{20}{c}}
1 \\
0 \\
1
\end{array}} \right] + \beta \left[ {\begin{array}{*{20}{c}}
0 \\
1 \\
1
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
1 \\
1 \\
1
\end{array}} \right]$ so that $\alpha=1,~\beta=1~\&~\alpha+\beta=1$
Now what?
Well we supposed that $\left[ {\begin{array}{*{20}{c}}
1 \\
1 \\
1
\end{array}} \right] = \alpha \left[ {\begin{array}{*{20}{c}}
1 \\
0 \\
1
\end{array}} \right] + \beta \left[ {\begin{array}{*{20}{c}}
0 \\
1 \\
1
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
\alpha \\
\beta \\
{\alpha + \beta }
\end{array}} \right] \Rightarrow \begin{array}{*{20}{r}}
{\alpha = 1} \\
{\beta = 1} \\
{\alpha + \beta = 1}
\end{array}$ That is a condiction.
So it is proof by contradiction,