1. ## Linear combination problem

"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.

2. ## Re: Linear combination problem

Originally Posted by MrJank
"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?

3. ## Re: Linear combination problem

Well, if a = 1 and b = 1, a + b = 1 is a false statement.

Is that the answer to the question? I didn't think it would be easy as that...

4. ## Re: Linear combination problem

Originally Posted by MrJank
Well, if a = 1 and b = 1, a + b = 1 is a false statement.
Is that the answer to the question? I didn't think it would be easy as that...
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,