Find the transformation matrix: F : <mi mathvariant="double-struck">R 3 <

kokoszzm 2022-06-30 Answered
Find the transformation matrix:
F : R 3 [ x ]   ] R 3 [ x ]
F ( v ) = d 2 v d v 2
Basis: 1 , x , x 2 , x 3 and R 3 [ x ] - the set of all third degree polynomials of variable x over R Assume that all coefficients of the polynomials are 1
The first thing that springs to my mind is to calculate this derivative by hand, and so we got
d 2 y d y 2 = 2 + 6 x
Now, we need to put these values - 2 and 6 in such a matrix that - when multiplied by the basis vector -will give us 2 + 6 x But there are many ways I can think of, for example
[ 0 0 2 0 0 0 0 6 0 0 0 0 0 0 0 0 ]
Or maybe
[ 2 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 ]
Because both of them, when multiplied by [ 1 x x 2 x 3 ]
Will give the correct answer. Thus, what is the correct way to solve this?
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Answers (1)

Arcatuert3u
Answered 2022-07-01 Author has 30 answers
If you pick an ordered basis for the domain and co domain then the order is fixed.
Pick the basis x 1 , x x , x x 2 , x x 3

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