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Let V be the span of the vectors (1,3,-1,2),(-1,2,-1,-4). Express V as the solution set of a homogenous linear system.

Mathematics
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I'm not quite sure I understand the question...
I was having the same issue. i dont even know where to start lol
it's asking to express the general form I suppose?\[A\vec x=\vec0\]and we have\[A=\left[\begin{matrix}1&-1\\3&2\\-1&-1\\2&-4\end{matrix}\right]\]and\[\vec x=\left[\begin{array}~x_1\\x_2\\x_3\\x_4\end{array}\right]\]but what does it want us to write exactly?

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Other answers:

oh that matrix is sideways I think
no, it's just that the \(\vec x\) is 2 rows long\[A\vec x=\vec0\]and we have\[A=\left[\begin{matrix}1&-1\\3&2\\-1&-1\\2&-4\end{matrix}\right]\]and\[\vec x=\left[\begin{array}~x_1\\x_2\end{array}\right]\]but what does it want us to write?
ummm i have no clue
well, the solution set is supposed to be the vector space, so we need to write the set of all solution in set notation somehow it seems
but it says the "homogeneous solution" so I guess let's first solve the above
but I'm thinking maybe homogeneous doesn't mean =0
ohhhh ya i think it does mean =0
so then I think that the span is the set\[V=\{\vec x:A\vec x=\vec0\}\]where\[A=\left[\begin{matrix}1&-1\\3&2\\-1&-1\\2&-4\end{matrix}\right]\]
not sure how exactly they want you to state it, but the set of all vectors that solve that system seems to be the vector space they are talking about
so its just saying to express the V as the solution set of a homogenious system?
usually the span of those vectors is the set of all linear combinations of those vectors, so you if not for the word "homogenous" I would say the span is just\[S=\{c_1\langle1,3,-1,2\rangle\},c_2\langle-1,2,-1,4\rangle\}\]
typo, I meant \[S=\{c_1\langle1,3,-1,2\rangle,c_2\langle-1,2,-1,4\rangle\}\]
hard to know what they want you to write exactly imo
correct i definitely see that
but I don't think what I just wrote is "homogenous"

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