Originally Posted by

**iarapovic** Hey Everyone,

Tomorro i have an exam Linear Algebra, and i have been given a mock exam to practice on. Unfortunatrly my lecturer decided not to print the solutions to the mock exam and i have no way if knowing what i am doing is right!! Can i please have some answers?

1.Define the terms eigenspace, generalised eigenspace for an operator on Rn.

2.Show that the dimension of the eigenspace cannot be greater than the dimension of the generalised eigenspace.

3. Define the terms nilpotent matrix, nilpotent operator

on a vector space.

4.By finding eigenvalues of the matrix

A = [4 -1]

[1 2]

show A is the sum of a nilpotent and a diagonal matrix, and exponentiate A. Using this result, write down the solution of the system of Ordinary

Differential Equations x' = Ax

5. Define the terms ring, field, polynomial of degree n over a field F, root of a polynomial, polynomial function, and finally zero of a polynomial function.

6.Define the terms homomorphism of groups,

isomorphism of groups. Define groups and subgroups.

7.Define the term onto, one-one and bijective for maps from a set A to a set B. Show that the composite of onto maps is onto!