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Math Help - Scalar/Vector problems

  1. #1
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    Scalar/Vector problems

    I got stuck on a few questions for a linear algebra assignment...

    1. Show that there exists scalars c1, c2, and c3, not all 0, such that c1(-2,9,6) + c2(-3,2,1) + c3(1,7,5) = (0,0,0)

    2. a) If v = (3,7) is a vector in the xy-coordinate system, what are the components of v in the x1y1-coordinate system?

    b) If v = (v1v2) is a vector in the xy-coordinate system, what are the components of v in the x1y1-coordinate system?

    This kind of math is new to me this year
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  2. #2
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    Hello chrisks

    Welcome to Math Help Forum!
    Quote Originally Posted by chrisks View Post
    I got stuck on a few questions for a linear algebra assignment...

    1. Show that there exists scalars c1, c2, and c3, not all 0, such that c1(-2,9,6) + c2(-3,2,1) + c3(1,7,5) = (0,0,0)

    2. a) If v = (3,7) is a vector in the xy-coordinate system, what are the components of v in the x1y1-coordinate system?

    b) If v = (v1v2) is a vector in the xy-coordinate system, what are the components of v in the x1y1-coordinate system?

    This kind of math is new to me this year
    For question 1, you can prove that the determinant \begin{vmatrix}<br />
-2&-3&1\\<br />
9&2&7\\<br />
6&1&5\\<br />
\end{vmatrix} is zero. This will prove that its rows are linearly dependent, and hence the scalars c_1, c_2, c_3 exist.

    Or you can do it directly with the following system of equations:
    \left\{\begin{array}{l l}<br />
-2c_1-3c_2+c_3=0&(1)\\<br />
9c_1+2c_2+7c_3=0&(2)\\<br />
6c_1+c_2+5c_3=0&(3)\\<br />
\end{array}\right.
    Multiply (1) by 3 and add to (3):
    -8c_2+8c_3=0

    \Rightarrow c_2=c_3
    Substitute into (1):
    -2c_1 -3c_2+c_2=0

    c_1=-c_2
    So if we put c_1 = 1 and c_2 = -1 = c_3, then:
    1\begin{pmatrix}-2\\9\\6\end{pmatrix}-1\begin{pmatrix}-3\\2\\1\end{pmatrix}-1\begin{pmatrix}1\\7\\5\end{pmatrix}=\begin{pmatri  x}0\\0\\0\end{pmatrix}
    For questions 2 and 3, I'm afraid I don't understand what
    x1y1-coordinate system
    means.

    Do you have any more information about this?

    Grandad
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  3. #3
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    Actually, I was able to figure out those last two problems, but thank you so much for your help on the first one! Much appreciated!
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