I am having trouble proving that the following are linear transformation:

1) T:R^3---> R^2 is an L.T defined by T(a1,a2,a3)= (a1-a2,2a3)

2)T:P2(R)--->R^3be a linear transformation defined by T(a0+a1x+a2x^2)=(a0,a1,a2)

thanks

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- Jun 20th 2008, 03:16 AMJCIRLinear Algebra please help.Exam in a couple of hours.
I am having trouble proving that the following are linear transformation:

1) T:R^3---> R^2 is an L.T defined by T(a1,a2,a3)= (a1-a2,2a3)

2)T:P2(R)--->R^3be a linear transformation defined by T(a0+a1x+a2x^2)=(a0,a1,a2)

thanks - Jun 20th 2008, 03:35 AMkalagota
$\displaystyle T((a,b,c) + (u,v,w)) = T(a+u, b+v, c+w) = (a+u-(b+v), 2(c+w))$

$\displaystyle

= ((a-b)+(u-v), 2c + 2w) = (a-b, 2c) + (u-v, 2w) = T(a,b,c) + T(u,v,w)$

$\displaystyle T(\alpha (a,b,c)) = T(\alpha a, \alpha b, \alpha c) = (\alpha a - \alpha b, 2\alpha c)$

$\displaystyle = \alpha (a-b, 2c) = \alpha T(a,b,c)$

just do it by definition.. you can easily do the second one..