1
$\begingroup$

I have done another questions considering spanning sets but I really don't know how to solve this one, I think I don't even totally get what is meant.

Any help is appreciated :)

$$x_1=\left[ \begin{matrix}-1\\2\\3\end{matrix} \right] \quad x_2=\left[ \begin{matrix}3\\4\\2\end{matrix} \right] \quad x =\left[ \begin{matrix}2\\6\\6\end{matrix} \right] \quad y =\left[ \begin{matrix}-9\\-2\\5\end{matrix} \right] $$

  1. is $x \in \operatorname{span}(x_1,x_2)$?

  2. is $y \in \operatorname{span}(x_1,x_2)$?

Start of my attempt (which might already be a failure):

$$x=\alpha_1x_1+\alpha_2x_2$$

so $$\left[ \begin{matrix} 2\\6\\6 \end{matrix} \right] = \alpha_1 \left[ \begin{matrix} -1\\2\\3 \end{matrix} \right] + \alpha_2 \left[ \begin{matrix}3\\4\\2\end{matrix} \right]$$

Well I dont know whether my approach is correct, nor how to continue

Thanks in advance

  • 0
    If you have problem typesetting vectors use $v = (v_1,\ldots,v_n)^ T$! – 2017-01-02

1 Answers 1

1

This is correct. Now you just need to write down the three equations this implies: $$-\alpha_1+3\alpha_2=2,\\ 2\alpha_1+4\alpha_2=6, \\ 3\alpha_1+2\alpha_2=6$$ and see if there exists a solution.

  • 0
    this results in $\alpha_2$ being both 1 and 12/11, so that means it is inconsistent and doesn't span x, right? – 2017-01-02
  • 0
    @Amaluena: Correct. – 2017-01-02