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In a linear system

$$x - 2y + 3z = 0 \\ -3x + 7y -8z = 0 \\ -2x + 4y -6z = 0 $$

The solution space is $\{(2s -3t,s,t) \mid s,t \in \mathbb{R}\} = \operatorname{span}\{(2,1,0),(-3,0,1)\}$

How do we know that the plane in $\mathbb{R}^3$ contains the origin?

hmakholm left over Monica
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TsTeaTime
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    Is (0,0,0) a solution to the system? It's easy to check. Plug in 0 for x, y and z and see if it makes all three equations true. – anon Jul 26 '16 at 11:10
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    This is what's called a "homogeneous system of equations" The origin always satisifes the system, just plug in zeros for all the variables and you'll see all equations hold true. – Gregory Grant Jul 26 '16 at 11:10
  • Are you asking how to know that the solution space that you give is correct, or how to know that the solution space that you give contains the origin? – Carsten S Jul 26 '16 at 17:03
  • You might want to check that solution space. [The intersection of two non-parallel planes (1st and 3rd equations are the same) is going to be a line.] – Hao Ye Jul 26 '16 at 18:52

2 Answers2

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The solution set contains the origin because if you plug $x=y=z=0$ into your equations, each of them becomes $0=0$ which is true.

If you have nonzero constant terms in one or more of your equations, this will not be the case, and you need to offset the span by a known particular solution, if there are any solutions at all.

hmakholm left over Monica
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This is what's called a "homogeneous system of equations" The origin always satisifes the system, just plug in zeros for all the variables and you'll see all equations hold true.

If you don't have zeros on the right hand side then it's harder, you have to row-reduce the augmented matrix of coefficients in that case.

Gregory Grant
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