# Algebra: endomorphism

1. Jan 28, 2008

### Godfrey

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:

a)

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. Jan 28, 2008

### psi^

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:
$$f(x_1, x_2, x_3) = (x_2 + x_3, x_1 + x_3, x_2 - x_1)$$

Hints:
a) multiply A by $$(x_1, x_2, x_3)$$ 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 $$(x_1, x_2, x_3)$$, check a)
c2) you get the kernel from knowing which vectors v get $$f(v) = \bar{0}$$, solve the system.