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Give algebraic and geometric descriptions of $\operatorname{Span} \{ a_1, a_2, a_3, a_4 \}$ where

$a_1 = (1, -1, -2), a_2 = (3, -3, -1), a_3 = (2, -2, -4), a_4 = (2, -2, 1)$

So far, I have: $ \begin{matrix} \;\;\,1 & \;\;\,3 & \;\;\,2 & \;\;\,2 \\ -1 &-3 &-2 &-2\\ -2 & -1 &-4 & \;\;\,1 \\ \end{matrix} $ Though, I feel like I'm missing a column. What should this system be equal to?

3 Answers 3

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$\begin{pmatrix} 1 & 3 & 2 & 2\\ -1& -3& -2& -2\\ -2& -2& 4& 1 \end{pmatrix}$

only use elementary row operation,we can get

$ \begin{pmatrix} 1 &0 &2 &-1 \\ 0&1 & 0 &1 \\ 0&0 &0 &0 \end{pmatrix}$

then,$a_1=2a_3$,and $a_4=-a_1+a_2$

$\operatorname{Span} \{ a_1, a_2, a_3, a_4 \}=\operatorname{Span} \{ a_1, a_2\}$

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    if you think my answer is right,you can select it as the best answer of your question.2012-09-10
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To start, notice that $a_1=2a_3$, thus we may eliminate $a_3$ as redundant. Can you see a relation between the remaining vectors?

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    I calculated the matrix of a2, a4 = a1 and got 1 0 | 3, 0 1 | -1, 0 0 | 0, but I'm not sure what this means exactly.2012-09-10
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For the geometric description, observe that all four vectors are linear combinations of $(1,-1,0)$ and $(0,0,1)$. So the span is the plane that contains $(0,0,0)$ and those two points. This plane includes the $Z$-axis, and makes $45^\circ$ angles with the $X$- and $Y$-axes.