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Math Help - How to show this is Linear? (Urgent!)

  1. #1
    lo2
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    How to show this is Linear? (Urgent!)

    I have this Linear Transformation f: P_3(R) \rightarrow R^2 given by

    f(P(x))=(P(1),P(-2))

    Then I have to show that f is linear.

    I do not really have any idea... SO really need some help please!
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  2. #2
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    Show that the following is true.
    f\left( {\alpha P(x) + Q(x)} \right) = \alpha f\left( {P(x)} \right) + f\left( {Q(x)} \right)
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  3. #3
    lo2
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    Quote Originally Posted by Plato View Post
    Show that the following is true.
    f\left( {\alpha P(x) + Q(x)} \right) = \alpha f\left( {P(x)} \right) + f\left( {Q(x)} \right)
    I can show that the right side is: k= \alpha

    k(P(1), P(-2))+(Q(1), Q(-2))

    But I do not really know what do with the left side...
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  4. #4
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    Quote Originally Posted by lo2 View Post
    I can show that the right side is: k= \alpha

    k(P(1), P(-2))+(Q(1), Q(-2))

    But I do not really know what do with the left side...
    That makes no sense at all! f is supposed to map into R^2 but you have 3 components! In R^2 addition is defined by (a,b)+ (c,d)= (a+c, b+d), "component wise".

    f is a function from P_3(R) which, I think, is the space of polynomials of degree 3 or less with real coefficients, to R2, the space of pairs of real numbers. f(P(x)) is defined as the pair (P(1), P(-2)).

    So, for example, if [tex]P(x)= x^3+ 2x- 1[/itex], P(1)= 1+ 2- 1= 2 and P(-2)= -8- 4-1= -13. f(P)= (2, -13).

    f(\alpha P+ Q)= (\alpha P(1)+ Q(1), \alpha P(2)+ Q(2)) and \alpha f(P)+ Q(P)= \alpha (P(1), P(2))+ (Q(1), Q(2). Are those the same?
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