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    Linear Transformations : Basis, Kernel, Image, Onto, One-to-one and Matrices

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    1) Define a linear transformation....
    a) Find a basis for Ker T.
    b) Find a basis for Im T.
    c) Is T an onto map?
    d) Is T a one-to-one map?

    2) Define a linear transformation...
    a) Find the matrix for T with respect to the standard basis.
    b) Find the matrix for T with respect to { ( ) , ( ) , ( ) } as the basis for R and the standard basis for R2.

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    © BrainMass Inc. brainmass.com May 20, 2020, 1:11 pm ad1c9bdddf
    https://brainmass.com/math/linear-transformation/linear-transformations-basis-kernel-image-onto-one-to-one-and-matrices-42424

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    (1) T(ax^2 + bx + c) = (a + b)x^2 + (b + c)x

    (a) Find a basis for Ker T

    Ker T is the set of all elements that are mapped into zero.
    So a + b = 0 and b + c = 0
    So b = -a and c = -b = a
    Ker T = { ax^2 - ax + a where a is any real number}

    I don't know what format you are using for basis; if you treat the coefficients as a vector, the basis is a single vector (1,-1,1) Only one ...

    Solution Summary

    Linear Transformations, Basis, Kernel, Image, Onto, One-to-one and Matrices are investigated.

    $2.19

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