# Linear Transformation and isomorphisms

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1. Aug 30, 2015

### Eleni

1. The problem statement, all variables and given/known data
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]
e) is T an isomorphism? Explain. If it is an isomorphism, find [T][/-1]
f) Find all polynomials in [p][/2] that solve T(p)= (5,0,-2)

2. Relevant equations
?

3. 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.

2. Aug 30, 2015

### andrewkirk

Your notation is unclear. Are you trying to write latex? If so you need to enclose it within delimiters, such as double-# to open and close the code for in-line latex and double-\$ to open and close for stand-alone 'display' latex.

But even with delimiters, the code you are writing doesn't look like correct latex.

So far as I can tell your function T maps from the module of polynomials of order two or less to the module (in fact vector space) $\mathbb{R}^3$.

Your answers to (a) and (b) look to take the correct approach (I didn't check results though), and I don't think you need to do any more on (b) than what you've written.

But from (c) onwards the notation becomes too hard to decipher. Can you try re-posting using proper latex? There's a primer here.