# Algebra: endomorphism

1. Jan 28, 2008

### Godfrey

1. The problem statement, all variables and given/known data

Be a base of . And be f a defined endomorphism by the expression :

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

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)

The b and c, how could I solve them?.

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.