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Thread: help with Linear spaces:)

  1. #1
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    help with Linear spaces:)

    Hey,
    Is the group a field in relation to the addition and multiplication of the matrices?
    help with Linear spaces:)-win_20180523_21_24_07_pro-2-.jpg
    Thanks
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  2. #2
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    Re: help with Linear spaces:)

    A basic property of a field is that every member of a field, other than the additive identitiy (0), has a multiplicative inverse. Such a matrix has a multiplicative inverse if and only if its determinant is $\displaystyle a^2- 2b^2\ne 0$. That is always true for a and b rational numbers.
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  3. #3
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    Re: help with Linear spaces:)

    Should I show the 7 features?
    How do I start it?
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  4. #4
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    Re: help with Linear spaces:)

    Can't you use the fact that
    1) addition of matrices is commutative.
    2) addition of matrices is associative.
    3) there exist a "0" matrix, the additive identity.
    4) every matrix has an additive inverse.
    5) multiplication of matrices is associative.
    6) there exist a "1" matrix, the multiplicative identity.
    are true of ALL matrices? so the only thing left to show is that the every non-zero matrix is invertible.

    (Well, I guess you should show that the 0 matrix, $\displaystyle \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$ is of the given form, with a= b= 0 and that the identity matrix, $\displaystyle \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$, is of that form with a= 1, b= 0.)
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  5. #5
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    Re: help with Linear spaces:)

    Quote Originally Posted by HallsofIvy View Post
    Can't you use the fact that
    1) addition of matrices is commutative.
    2) addition of matrices is associative.
    3) there exist a "0" matrix, the additive identity.
    4) every matrix has an additive inverse.
    5) multiplication of matrices is associative.
    6) there exist a "1" matrix, the multiplicative identity.
    are true of ALL matrices? so the only thing left to show is that the every non-zero matrix is invertible.

    (Well, I guess you should show that the 0 matrix, $\displaystyle \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$ is of the given form, with a= b= 0 and that the identity matrix, $\displaystyle \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$, is of that form with a= 1, b= 0.)

    That's the solution?
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