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1. Find a linear map from P

_{4}to P

_{4}(where P

_{4}is the space of polynomials of degree less than 4) whose kernel is one-dimensional. Find one whose kernel is two-dimensional.

what im thinking is:

if it is a map in P

_{4}, then it looks like

[tex]

\left|\begin{array}{ccc}1 \\ x\\ x^2\\x^3\end{array}\right|

[/tex]

?? not sure where to go from there.

2. Find the matrix representation in the standard basis of the linear transformation from P

_{4}to P

_{3}:

*f*(

*x*) -> 2

*f*(

*x*) -

*f*(

*x*-1) -

*f*(

*x*+1).

What is the dimension of its kernel? Of its image?

absolutely no idea where to begin