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Here are the linear equation: $$2x+z=0$$ $$-x+3y+z=0$$ $$-x+y+z=0$$

I have found that the general solution is, $$t \begin{bmatrix} \frac{-1}2\\ \frac{-1}2 \\ 1 \\ \end{bmatrix} $$

The question asks me to find the geometric interpretation of general solution. But I have no idea how. I think my limit knowledge of geometric interpretation is not helping me, so some explanation about geometric interpretation would help me a lot. Thanks you for any help!

Chow
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2 Answers2

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What you have is the parametric equation for a line. Do you see why?

Potato
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  • Sorry, I don't. Doesn't need to be square for parametric equation? – Chow Nov 07 '14 at 05:09
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    @Chow What do you think needs to be square? Try to graph something like $t[1,2]$ in the $(x,y)$ plane to get a feel for what these kinds of graphs look like. – Potato Nov 07 '14 at 05:12
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Your solution:

$$t \begin{bmatrix} \frac{-1}2\\ \frac{-1}2 \\ 1 \\ \end{bmatrix} $$

represents every multiple of the single vector $[-1/2,-1/2,1]^T$. That one vector represents a point in $\mathbb{R}^3$. Think of it, and all of its multiples, as a set of points in $\mathbb{R}^3$. Positive multiples (positive values of $t$) are on the same side of the origin as the original vector; negative multiples are on the opposite side of the origin. The zero-multiple (taking $t=0$) is the origin.

Do you see how this is a description of a line through the origin?

G Tony Jacobs
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