Explore BrainMass
Share

Explore BrainMass

    Linear transformation in Matrix form

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    Let P3 denote the real vector space of polynomial functions of degree up to 3, i.e.

    p3-{f(x)=a3x^3+a2x^2+a1x| aiER}

    Consider the linear transformation D: P3--> P3 given by the derivative D(f) = d/dx f
    a) What is the kernel of D? Give a basis of the kernel
    b) The set B={1, x, x^2, x^3} forms a basis of D. Write the matrix Mb^b (D)
    c) What is the rank of D
    d) Show that the 4-fold composition D^4 = D*D*D*D (i.e. applying D to the vector space four times) is the zero map.

    © BrainMass Inc. brainmass.com October 10, 2019, 7:47 am ad1c9bdddf
    https://brainmass.com/math/linear-algebra/linear-transformation-matrix-form-594944

    Attachments

    Solution Preview

    Hello and thank you for posting your question to Brainmass.
    The solution ...

    Solution Summary

    The solution shows in detail, how to find the matrix representation of the derivative transformation in the basis of polynomials of degree 3, how to find its kernel and rank and showing that applied 4 times it indeed becomes the zero map.

    $2.19