linear transformation?

Linear Transformation T(p(x))=p(2x+1)-8p(x) Find image of T since span{1 x x^2 x^3}

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  • 9 years ago
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    In general, T(x^n) = (2x + 1)^n - 6 x^n = (2^n - 8)x^n + polynomial of lower degree

    T(x^0) = - 7x^0 so x^0 = 1 is in the image of (T)

    From the general formula, if n is not 3 (so that 2^n - 8 is non-0)

    then if all polynomials of degree < n are in Im(T), so is x^n

    Therefore 1, x, and x^2 are in the image of T. However, x^3 is not.

    Image pf T = span(1, x, x^2) = all polynomials of degree 2

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