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Math Help - Some Linear Algebra

  1. #1
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    Some Linear Algebra

    For the following, show that  M_{m\times n}(\mathbb{R}) does not form a vector space over  \mathbb{R} under the (non standard) operations  \hat{+} (addition) and * (multiplication).

    (a)  A_{ij} \ \hat{+} \ B_{ij} = A_{ij} + B_{ij}
     k * A_{ij} = k^2A_{ij}

    (b)  A_{ij} \ \hat{+} \ B_{ij} = A_{ij}B_{ij}
     k * A_{ij} = {A^k}_{ij}
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  2. #2
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    Quote Originally Posted by funnyinga View Post
    For the following, show that  M_{m\times n}(\mathbb{R}) does not form a vector space over  \mathbb{R} under the (non standard) operations  \hat{+} (addition) and * (multiplication).

    (a)  A_{ij} \ \hat{+} \ B_{ij} = A_{ij} + B_{ij}
     k * A_{ij} = k^2A_{ij}
    multiplication by scalar is obviously not linear.

    (b)  A_{ij} \ \hat{+} \ B_{ij} = A_{ij}B_{ij}
     k * A_{ij} = {A^k}_{ij}
    the additive inverse doesn't always exist! see that the additive identity, namely 0, here is the matrix with all entries equal 1. now let A be the matrix with 0 in the first row and first column and

    whatever you want everywhere else. then A has no additive inverse.
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