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Thread: Linear Algebra Vector Question

  1. #1
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    Linear Algebra Vector Question

    1. In the following case, determine whether or not u is in the plane spanned by v and w.

    u = $\displaystyle \left(\begin{array}{cc}13\\7\end{array}\right)$ , v = $\displaystyle \left(\begin{array}{cc}-5\\6\end{array}\right)$ , w = $\displaystyle \left(\begin{array}{cc}4\\9\end{array}\right)$

    What I was thinking I had to do was do the following:
    $\displaystyle \left(\begin{array}{cc}x\\y\end{array}\right)$ = a$\displaystyle \left(\begin{array}{cc}-5\\6\end{array}\right)$ + b$\displaystyle \left(\begin{array}{cc}4\\9\end{array}\right)$

    and then from here I would solve for a and b but I have no idea if it's right

    Thanks in advance to whoever helps!
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  2. #2
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    Re: Linear Algebra Vector Question

    Quote Originally Posted by qzno View Post
    1. In the following case, determine whether or not u is in the plane spanned by v and w.

    u = $\displaystyle \left(\begin{array}{cc}13\\7\end{array}\right)$ , v = $\displaystyle \left(\begin{array}{cc}-5\\6\end{array}\right)$ , w = $\displaystyle \left(\begin{array}{cc}4\\9\end{array}\right)$

    What I was thinking I had to do was do the following:
    $\displaystyle \left(\begin{array}{cc}x\\y\end{array}\right)$ = a$\displaystyle \left(\begin{array}{cc}-5\\6\end{array}\right)$ + b$\displaystyle \left(\begin{array}{cc}4\\9\end{array}\right)$

    and then from here I would solve for a and b but I have no idea if it's right

    Thanks in advance to whoever helps!
    One question: Is it true $\displaystyle \left| {\begin{array}{rl} { - 5} & 4 \\ 6 & 9 \\ \end{array} } \right| \ne 0\;?$

    AND why is that question significant?
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  3. #3
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    Re: Linear Algebra Vector Question

    i dont really understand what you are asking Plato
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  4. #4
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    Re: Linear Algebra Vector Question

    Quote Originally Posted by qzno View Post
    i dont really understand what you are asking Plato
    That may well why you have trouble with the question in the first place.
    The question I asked you has everything to do with if this system has a unique system.
    $\displaystyle \left\{ \begin{gathered} - 5a + 4b = x \hfill \\ 6a + 9b = y \hfill \\ \end{gathered} \right.$

    Does that system have a unique solution?
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  5. #5
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    Re: Linear Algebra Vector Question

    i solved for a and b and got really weird numbers so im guessing it doesnt have one.?
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  6. #6
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    Re: Linear Algebra Vector Question

    Quote Originally Posted by qzno View Post
    i solved for a and b and got really weird numbers so im guessing it doesnt have one.?
    Are you really studying linear algebra?
    Does $\displaystyle \left[ {\begin{array}{*{20}c} { - 5} & 4 \\ 6 & 9 \\ \end{array} } \right]^{ - 1} $ exist?

    If so, then there is a unique solution.

    You may need to seek live help from your instructor.
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  7. #7
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    Re: Linear Algebra Vector Question

    [QUOTE=qzno;680875]1. In the following case, determine whether or not u is in the plane spanned by v and w.

    u = $\displaystyle \left(\begin{array}{cc}13\\7\end{array}\right)$ , v = $\displaystyle \left(\begin{array}{cc}-5\\6\end{array}\right)$ , w = $\displaystyle \left(\begin{array}{cc}4\\9\end{array}\right)$

    What I was thinking I had to do was do the following:
    $\displaystyle \left(\begin{array}{cc}x\\y\end{array}\right)$ = a$\displaystyle \left(\begin{array}{cc}-5\\6\end{array}\right)$ + b$\displaystyle \left(\begin{array}{cc}4\\9\end{array}\right)$[quote]
    I see no reason to use "x" and "y" when you are given specific components for u.
    Why not $\displaystyle \begin{pmatrix}13 \\ 7\end{pmatrix}= a\begin{pmatrix}-5 \\ 6 \end{pmatrix}+ b\begin{pmatrix} 4 \\ 9 \end{pmatrix}$
    which gives you two equations for a and b.

    and then from here I would solve for a and b but I have no idea if it's right

    Thanks in advance to whoever helps!
    By the way, using "\begin{pmatrix}... \end{pmatrix} is a little easier than "\(\begin{array}{cc}... \end{array}\right)"! The "p" is for "parentheses" \begin{bmatrix} gives brackets:

    \begin{pmatrix} 4 \\ 9\end{pmatrix}:
    $\displaystyle \begin{pmatrix} 4 \\ 9 \end{pmatrix}$

    \begin{matrix} 4 \\ 9 \end{pmatrix}:
    $\displaystyle \begin{bmatrix} 4 \\ 9 \end{bmatrix}$
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