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

1) T(M) = P M P^(-1), where P = [2 3; 5 7], fromR^(2x2) toR^(2x2)

2) T(f(t)) = [f(7); f(11)] fromP2 toR^2

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- October 13th 2009, 09:26 PMnoles2188Linear 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*P*2 to**R**^2 - October 14th 2009, 05:17 AMHallsofIvy
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?