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Math Help - Linear independence of matrix set?

  1. #1
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    Linear independence of matrix set?

    I need to show that the following set of 2x2 matrices forms a basis for the subspace that it spans, ie) that it is linearly independent.

    {[1 1] [0 1] [2 0] }
    {[1 0], [1 1], [0 -1]}

    For a vector set I would just form a columnspace matrix, row reduce, and ensure that there are no parameters (that the rank is equal to the amount of variables).

    How do I go about forming a matrix to use this method with the given 2x2 matrices.
    Or, what method should I use to show that this matrix set is linearly independent??
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  2. #2
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    Quote Originally Posted by crymorenoobs View Post
    I need to show that the following set of 2x2 matrices forms a basis for the subspace that it spans, ie) that it is linearly independent.

    {[1 1] [0 1] [2 0] }
    {[1 0], [1 1], [0 -1]}

    For a vector set I would just form a columnspace matrix, row reduce, and ensure that there are no parameters (that the rank is equal to the amount of variables).

    How do I go about forming a matrix to use this method with the given 2x2 matrices.
    Or, what method should I use to show that this matrix set is linearly independent??

    Suppose a\begin{pmatrix}1&1\\1&0\end{pmatrix}+b\begin{pmat  rix}0&1\\1&1\end{pmatrix}+c\begin{pmatrix}2&0\\0&\  !\!\!-1\end{pmatrix}=\begin{pmatrix}0&0\\0&0\end{pmatrix  } , a,b,c\in \mathbb{R} (in case you meant real matrices), then:

    \begin{pmatrix}a+2c&a+b\\a+b&b-c\end{pmatrix}=\begin{pmatrix}0&0\\0&0\end{pmatrix  }

    Well, now just compare entry-entry in both sides and conclude that the only solution is a=b=c=0\Longrightarrow your matrices are lin. ind.

    Tonio
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  3. #3
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    Ah kk thanks.
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