Question 8896: 1; let v be the space of all polynomials over a field F of degree less or equal to 3
let D be a diff. operator from v to v
1; find the matrix A of D with respect to the basis {1,,x,x.x,x.x.x} of v .
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! With your level,it seems that you should know how to type
{1,x,x^2,x^3} instead of the ugly form you have given.
Let D: V-->V (capital in general. It is important to follow some convention
in math.)
Any linear transformation is unquely determined by its values on the
basis.
Now, D(1) =0, (corresponding to the 1st column of A)
D(x) = 1 , (the 2nd column of A)
D(x^2) = 2x = 2 * x, (the 3rd column of A) and
D(x^3) = 3x^2 = 3* x^2. (the 4th column of A)
Hence, the matrix A od D wrtto the basis {1,x,x^2,x^3}
is
[0 1 0 0
0 0 2 0
0 0 0 3
0 0 0 0 ]
Note: D(xi) = E Aik xk (E means summation k from 1 to 3
{xi} is a basis)
If you have trouble understanding,try to review the def. about
the matrix representaion for a linear transformation carefully.
After all, this is a very basic problem in linear algebra.
Kenny
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