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Math Help - Linear Transformation/Isomorphism

  1. #1
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    Linear Transformation/Isomorphism

    Are these transformations linear? If so, are they isomorphisms?

    1) T(M) = P M P^(-1), where P = [2 3; 5 7], from R^(2x2) to R^(2x2)

    2) T(f(t)) = [f(7); f(11)] from P2 to R^2
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  2. #2
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    Quote Originally Posted by noles2188 View Post
    Are these transformations linear? If so, are they isomorphisms?

    1) T(M) = P M P^(-1), where P = [2 3; 5 7], from R^(2x2) to R^(2x2)

    2) T(f(t)) = [f(7); f(11)] from P2 to R^2
    What have you tried? The definition of "linear transformation" is, of course, that f(au+ bv)= af(u)+ bf(v) for any scalars a and b and vectors u and v. Here the "vectors" are two by two matrices.

    Is P(aM+ bN)P^(-1)= aPMP^(-1)+ bPNP^{-1}?

    Is T(af(t)+ bg(t))= aT(f(t))+ bT(g(t))?

    A linear transformation is an isomorphism if and only if it is one-to-one.

    If T(M)= T(N), does it follow that M= N?

    If T(f(t))= T(g(t)), does it follow that f= g?
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