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

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.

Please see the attached file for the fully formatted problems.

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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.

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