Linear transformations

  • Thread starter baddin
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  • #1
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1. Give information
Let T: P3 ---> P3 be the linear transformation described by:
T(p(x))=p(x+1)+p(2-x).
Find the matrix of T with respect to the standard basis b {1,x,x^2,x^3}.


The Attempt at a Solution


I found the transformations on the standard basis b:
T(1) = 2
T(x) = 3
T(x^2) = 2x^2 -2x +5
T(x^3) = 9x^2 - 9x + 9
I am confused on what to do next...
 

Answers and Replies

  • #2
Dick
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1. Give information
Let T: P3 ---> P3 be the linear transformation described by:
T(p(x))=p(x+1)+p(2-x).
Find the matrix of T with respect to the standard basis b {1,x,x^2,x^3}.


The Attempt at a Solution


I found the transformations on the standard basis b:
T(1) = 2
T(x) = 3
T(x^2) = 2x^2 -2x +5
T(x^3) = 9x^2 - 9x + 9
I am confused on what to do next...

Write your functions so they look a little more like vectors, write a+bx+cx^2+dx^3 as the column vector [a,b,c,d]. So T(1)=2 becomes T([1,0,0,0])=[2,0,0,0]. Does that help?
 
  • #3
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Ok, then I should find T(1,0,0,0), T(0,1,0,0), T(0,0,1,0) and T(0,0,0,1) right?
 
  • #4
Dick
Science Advisor
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26,260
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Ok, then I should find T(1,0,0,0), T(0,1,0,0), T(0,0,1,0) and T(0,0,0,1) right?

Right. You really already did. Just write them as column vectors. Then those will be the columns of your matrix.
 
  • #5
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Okay thank you very much for your help =)
 

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