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BenBlackburn

  • 2 years ago

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.

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  1. colorful
    • 2 years ago
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    I'm not quite sure I understand the question...

  2. BenBlackburn
    • 2 years ago
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    I was having the same issue. i dont even know where to start lol

  3. colorful
    • 2 years ago
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    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?

  4. colorful
    • 2 years ago
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    oh that matrix is sideways I think

  5. colorful
    • 2 years ago
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    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?

  6. BenBlackburn
    • 2 years ago
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    ummm i have no clue

  7. colorful
    • 2 years ago
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    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

  8. colorful
    • 2 years ago
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    but it says the "homogeneous solution" so I guess let's first solve the above

  9. colorful
    • 2 years ago
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    but I'm thinking maybe homogeneous doesn't mean =0

  10. BenBlackburn
    • 2 years ago
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    ohhhh ya i think it does mean =0

  11. colorful
    • 2 years ago
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    so then I think that the span is the set\[V=\{\vec x:A\vec x=\vec0\}\]where\[A=\left[\begin{matrix}1&-1\\3&2\\-1&-1\\2&-4\end{matrix}\right]\]

  12. colorful
    • 2 years ago
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    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

  13. BenBlackburn
    • 2 years ago
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    so its just saying to express the V as the solution set of a homogenious system?

  14. colorful
    • 2 years ago
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    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\[S=\{c_1\langle1,3,-1,2\rangle\},c_2\langle-1,2,-1,4\rangle\}\]

  15. colorful
    • 2 years ago
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    typo, I meant \[S=\{c_1\langle1,3,-1,2\rangle,c_2\langle-1,2,-1,4\rangle\}\]

  16. colorful
    • 2 years ago
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    hard to know what they want you to write exactly imo

  17. BenBlackburn
    • 2 years ago
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    correct i definitely see that

  18. colorful
    • 2 years ago
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    but I don't think what I just wrote is "homogenous"

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