- #1

Kernul

- 211

- 7

## Homework Statement

Being ##f : \mathbb R^4\rightarrow\mathbb R^4## the endomorphism defined by:

$$f((x,y,z,t)) = (13x + y - 2z + 3t, 10y, 9z + 6t, 6z + 4t)$$

1) Determine the basis and dimension of ##Ker(f)## and ##Im(f)##. Complete the base chosen in ##Ker(f)## into a base of ##\mathbb R^4##;

2) Determine the basis and dimension of the subspaces of ##\mathbb R^4 Ker(f) \cap Im(f)## and ##Ker(f) + Im(f)##;

3) Determine the preimage of the vector ##\vec v = (0, 2h, 1, h - 4)##, with the variation of the real parameter ##h##;

4) Find, if they exist, the values of the ##h## parameter with ##[2\vec e_1 + \vec e_2, 3\vec e_3, \vec v, -\vec e_2 + \vec e_4]## being a base of ##\mathbb R^4##. ##ε = (\vec e_1, \vec e_2, \vec e_3, \vec e_4)## is the canonic base of ##\mathbb R^4##.

5) Having with ##A## the matrix ##M_{εε}(f)## associated to ##f## in respect to the canonic base, find out if ##A## is diagonalizable and, if so, determine a diagonalizing (?) matrix ##P## and the corresponding diagonal matrix ##D## to which ##A## is similar.

6) Find out if the matrix ##B = \begin{pmatrix} 3 & 0 & 0\\ 1 & 3 & 0\\ 1 & 0 & 1\end{pmatrix}## is diagonalizable and show a maximum independent system of eigenvectors of ##B##

## Homework Equations

Rank-Nullity Theorem

Operation on the vector subspaces (intersection and sum)

Canonic Base

Diagonalization

Eigenvectors and eigenvalues

## The Attempt at a Solution

So, I know how to do the first point of the exercise but I found myself stuck.

In order to find the dimension and basis of both ##Ker(f)## and ##Im(f)## I first have to find the rank of the matrix $$A = \begin{pmatrix} 13 & 1 & -2 & 3\\ 0 & 10 & 0 & 0\\ 0 & 0 & 9 & 6\\ 0 & 0 & 6 & 4 \end{pmatrix}$$

From what I see, the matrix' rank is 4. Knowing that ##ρ(A) = dim(Im(f))##, the base of ##Im(f)## is made of all the columns of ##A##. So: $$B_{Im(f)} = [A^1, A^2, A^3, A^4]$$

Now, knowing that ##dim(Ker(f)) = dim(A) - dim(Im(f))##, I have ##dim(Ker(f)) = 4 - 4 = 0##. This means that ##Ker(f)## has no basis, right? Then how do I complete the basis of ##Ker(f)##(as the exercise asks) if there is none?

For now I would like to concentrate on this first point and then ask other questions for the following points, if needed.