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Homework Help: Algebra: endomorphism

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

    Be [PLAIN]http://www.rinconmatematico.com/latexrender/pictures/c2360a03b1cf0052b79abfea8051d3da.png [Broken] a base of http://www.rinconmatematico.com/latexrender/pictures/04065df7a06e6b7aedfd0b0519dd0736.png [Broken].[/URL] And be f a defined endomorphism by the expression [PLAIN]http://www.rinconmatematico.com/latexrender/pictures/415bffa5b3f9433f176071c3ce93a0ec.png: [Broken]

    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

    [PLAIN]http://www.rinconmatematico.com/latexrender/pictures/c2360a03b1cf0052b79abfea8051d3da.png [Broken]
    http://www.rinconmatematico.com/latexrender/pictures/415bffa5b3f9433f176071c3ce93a0ec.png [Broken]

    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:


    http://www.rinconmatematico.com/latexrender/pictures/6e396661515df828a2c2316f129c683d.png [Broken]

    The b and c, how could I solve them?.
    Last edited by a moderator: May 3, 2017
  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]

    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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