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Math Help - Matrix 2x2

  1. #1
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    Matrix 2x2

    If A = [a b]
    c d
    is a 2 x 2 matrix such that the vectors [a c] and [b d]


    are linearly independent, prove carefully that rank A = 2. (You cannot choose thematrix A - your proof must work for every 2 2 matrix with the property above,i.e. every 2 2 matrix with independent columns.)
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  2. #2
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    Re: Matrix 2x2

    Essentially, you want to show that \begin{bmatrix}a \\ c\end{bmatrix} and \begin{bmatrix}b \\ d\end{bmatrix} are linearly independent if and only if \begin{bmatrix}a & b\end{bmatrix} and \begin{bmatrix}c & d\end{bmatrix} are linearly independent.
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  3. #3
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    Re: Matrix 2x2

    yea that is right. but how to do that?
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  4. #4
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    Re: Matrix 2x2

    Assume that any linear combination e\begin{bmatrix}a \\ c\end{bmatrix}+f\begin{bmatrix}b \\ d\end{bmatrix} \neq \begin{bmatrix}0 \\ 0\end{bmatrix} unless e=f=0. Then show that any linear combination e\begin{bmatrix}a & b\end{bmatrix}+f\begin{bmatrix}c & d\end{bmatrix} \neq \begin{bmatrix}0 & 0\end{bmatrix} unless e=f=0.
    Last edited by SlipEternal; October 30th 2013 at 07:34 AM.
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