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Math Help - Trying to Prove three 2x2 Matrices are Linearly Independent

  1. #1
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    Question Trying to Prove three 2x2 Matrices are Linearly Independent

    Hey,

    Ive got a question asking;
    The set of M2,2 of all 2x2 matrices, with real entries, is a vector space. For what values of A (real) is the set
    Z= {(
    1 2
    1 0
    ,
    3 7
    0 0
    ,
    2 6
    A 0
    }

    a linearly independent subset of M2,2?

    I have been trying to prove using a1V1+a2v2+a3v3=0 but I cant figure how to prove it or put it into a singular matrix form for row reduction... any help would be greatly appreciated.
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  2. #2
    MHF Contributor

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    Re: Trying to Prove three 2x2 Matrices are Linearly Independent

    So you have a_1\begin{bmatrix}1 & 2 \\ 1 & 0 \end{bmatrix}+ a_2\begin{bmatrix}3 & 7 \\ 0 & 0 \end{bmatrix}+ a_3\begin{bmatrix}2 & 6 \\ A & 0 \end{bmatrix}
    = \begin{bmatrix}a_1+ 3a_2+ 2a_3 & 2a_1+ 7a_2+ 6a_3 \\ a_1+ Aa_3 & 0\end{bmatrix}= \begin{bmatrix}0 & 0 \\ 0 & 0 \end{bmatrix}
    so we must have a_1+ 3a_2+ 2a_3= 0, 2a_1+ 7a_2+ 6a_3= 0, and a_1+ Aa_3= 0 only if a_1+ a_2+ a_3= 0.

    Okay, have you tried solving for a_1, a_2, and a_3? If you subtract twice the first equation from the second, you eliminate a_1- a_2+ 2a_3= 0. If you subtract the third equation from the first you again eliminate a_1- 3a_2+ (2- A)a_3= 0. Now eliminate a_2 by subtracting twice the first of those two equations from the other: (2- A)a_3- 4a_3= (-2- A)a_3= 0.

    If a_3= 0 it follows that a_2= 0 and a_1= 0. What is the only value of A such that a_3 does not have to be 0?
    Last edited by HallsofIvy; March 27th 2013 at 06:37 AM.
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