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Algebra: endomorphism

  1. Jan 28, 2008 #1
    1. The problem statement, all variables and given/known data

    Be [​IMG] a base of [​IMG]. And be f a defined endomorphism by the expression [​IMG]:

    a) Identify the pooled?/associate? matrix refered to the base B.
    b) Identify the invariant vectors of f.
    c) Distinguish the kernel and the image in parametric and cartesian way.

    2. Relevant equations

    [​IMG]
    [​IMG]

    3. The attempt at a solution

    The paragraph a is the only one that I have some idea of how to solve it, and I made a solution, although I do not know if it is right:

    a)

    [​IMG]


    The b and c, how could I solve them?.
     
  2. jcsd
  3. Jan 28, 2008 #2
    The vector input in f is linear combination of the basis (and so is the output vector because its an endomorphism).
    think f like this:
    [tex] f(x_1, x_2, x_3) = (x_2 + x_3, x_1 + x_3, x_2 - x_1) [/tex]

    Hints:
    a) multiply A by [tex](x_1, x_2, x_3)[/tex] and verify results
    b) if Av = v then v has to be a combination of the columns of A
    c1) you get the image from the muliplying A by a input, say [tex](x_1, x_2, x_3)[/tex], check a) :rolleyes:
    c2) you get the kernel from knowing which vectors v get [tex]f(v) = \bar{0}[/tex], solve the system.
     
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