# 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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#### Solution Preview

(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.