- #1
Eleni
- 14
- 0
Homework Statement
Suppose a linear transformation T: [P][/2]→[R][/3] is defined by
T(1+x)= (1,3,1), T(1-x)= (-1,1,1) and T(1-[x][/2])=(-1,2,0)
a) use the given values of T and linearity properties to find T(1), T(x) and T([x][/2])
b) Find the matrix representation of T (relative to standard bases)
c) Let B={1,x,[x][/2]} and p=3-x-2[x][/2]. Find [p][/B]
d) Find T(3-x-2[x][/2]) in two different ways.
- directly using part (a) and the linearity of T, and
- calculating [T][[P]][/B]
f) Find all polynomials in [p][/2] that solve T(p)= (5,0,-2)
Homework Equations
?
The Attempt at a Solution
a) T(1)= (1)(1,3,1) + (1)(-1, 1, 1) +(1)(-1,2,0)
= (1,3,1)+(-1,1,1)+(-1,2,0)
= (-1,2,2)
T(x) = (1)(1,3,1) + (-1)(-1, 1, 1) +(0)(-1,2,0)
= (1,3,1)+(1,-1,-1)+(0,0,0)
= (2,2,0)
T([x][/2])= (0)(1,3,1) + (0)(-1, 1, 1) +(-1)(-1,2,0)
= (0,0,0)+(0,0,0)+(1,-2,0)
= (1,-2,0)
b) Since from part (a) The matrix representation of T relative to the standard bases is;
-1 2 1
2 2 -2
2 0 0
(I feel like I need to do more here but I'm not sure what).
c)Let B={1,x,[x][/2]} and p= 3-x+2[x][/2].
[p][/B] =(3,-1, 2)
(again I feel as if this was too simple and something is missing)
d) Find T(3-x-2[x][/2]) directly;
from part (a)
T(1)= (-1,2,2)
T(x)= (2,2,0)
T([x][/2])= (1,-1,0)
So T(3x-x+2[x][/2]) = (3)(-1,2,2) +(-1)(2,2,0)+(2)(1,-1,0)
= (-3,6,6)+(-2,-2,0)+(2,-2,0)
= (-3,2,6)
Calculating [T][[P]][/B];
-1 2 1 3 = -3
2 2 -1 -1 2
2 0 0 2 6
[T] [[P]][/B] =[T][[P]][/B]
I don't know where to begin with e) and f) they both look like the should be quite straight forward and just follow a formula or apply a theorem but I don't know what it would be. Any revisions to the work I have done so far are thoroughly appreciated and advice on how to approach and complete the parts I haven't attempted are warmly welcomed too. Thank you.