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Math Help - Linear Algebra Question.

  1. #1
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    Linear Algebra Question.

    Let A and B be vector spaces, (which turn out to be real btw) that have ordered bases alpha=(a1,a2,a3) and beta = (b1,b2,b3,b4). The matrix T: A->B is: 2 9 7 4 5 7 9 4 3 2 8 3 work out the alpha coord of a1+a2+a3 and hence or otherwise find T(a1+a2+a3) relative to b1,b2,b3,b4
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  2. #2
    Junior Member bleesdan's Avatar
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    OK, I'm a little confused about this.

    We have vector spaces A,B, with bases \alpha=(a_{1},a_{2},a_{3}) and \beta=(b_{1},b_{2},b_{3},b_{4}). We then have a matrix T= \left(\begin{array}<br />
2 & 9 & 7\\<br />
4 & 5 & 7\\<br />
9 & 4 & 3\\<br />
2 & 8 & 3<br />
\end{array}\left)<br />
    The matrix isn't working in the Latex editor, so let's just go on.

    OK.
    Tell me everything you know about vector space bases.
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  3. #3
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    The alpha coordinates (\alpha_1,\alpha_2,\alpha_3) represent the vector \alpha_1a_1+\alpha_2a_2+\alpha_3a_3

    Can you do the first part now?
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  4. #4
    Lord of certain Rings
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    Quote Originally Posted by SilenceInShadows View Post
    Let A and B be vector spaces, (which turn out to be real btw) that have ordered bases alpha=(a1,a2,a3) and beta = (b1,b2,b3,b4). The matrix T: A->B is: 2 9 7 4 5 7 9 4 3 2 8 3 work out the alpha coord of a1+a2+a3 and hence or otherwise find T(a1+a2+a3) relative to b1,b2,b3,b4
    Well since your T \equiv \begin{pmatrix}<br />
2 & 9 & 7\\<br />
4 & 5 & 7\\<br />
9 & 4 & 3\\<br />
2 & 8 & 3<br />
\end{pmatrix} and in terms of co-efficients a_1 + a_2 +a_3 translates to (1,1,1), finding T(a_1+a_2+a_3) amounts to computing \begin{pmatrix}<br />
 2 & 9 & 7\\<br />
 4 & 5 & 7\\<br />
 9 & 4 & 3\\<br />
2 & 8 & 3<br />
 \end{pmatrix}\begin{pmatrix}<br />
  1\\<br />
  1\\<br />
  1<br />
\end{pmatrix}


    Finally this will give you a 4 x 1 vector. These are the coefficients of b_1,b_2,b_3,b_4 in the linear combination of the vector in that order.

    Quote Originally Posted by bleesdan View Post
    OK, I'm a little confused about this.

    We have vector spaces A,B, with bases \alpha=(a_{1},a_{2},a_{3}) and \beta=(b_{1},b_{2},b_{3},b_{4}). We then have a matrix T= \left(\begin{array}<br />
2 & 9 & 7\\<br />
4 & 5 & 7\\<br />
9 & 4 & 3\\<br />
2 & 8 & 3<br />
\end{array}\left)<br />
    The matrix isn't working in the Latex editor, so let's just go on.

    OK.
    Tell me everything you know about vector space bases.
    Hey bleesdan,
    Use \begin{pmatrix}2 & 9 & 7\\4 & 5 & 7\\9 & 4 & 3\\2 & 8 & 3\end{pmatrix} to get \begin{pmatrix}<br />
2 & 9 & 7\\<br />
4 & 5 & 7\\<br />
9 & 4 & 3\\<br />
2 & 8 & 3<br />
\end{pmatrix}
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  5. #5
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    Thanks for your help, i have further questions; its most whats being asked of me that I cant figure.

    Ive been asked to write down the change matrix been beta to alpha. Where beta is: (4i+6j, 7i+2j, 3i+5j+9k) and alpha has the standard basis. Am I just being stupid and the change matrix is?

    4 6 0
    7 2 0
    3 5 9

    or do I need to do something to it? or?

    next part is find the matrix which represents
    4 5 6
    7 3 5
    3 5 8
    relative to beta
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  6. #6
    Senior Member
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    Ive been asked to write down the change matrix been beta to alpha. Where beta is: (4i+6j, 7i+2j, 3i+5j+9k) and alpha has the standard basis. Am I just being stupid and the change matrix is?

    4 6 0
    7 2 0
    3 5 9
    correct. Well done.

    next part is find the matrix which represents
    4 5 6
    7 3 5
    3 5 8
    relative to beta
    I'm not sure what this part is asking, sorry.
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