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Linear Transformation and isomorphisms

  1. Aug 30, 2015 #1
    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. jcsd
  3. Aug 30, 2015 #2

    andrewkirk

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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.
     
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