Review for midterm 2:

The test will be held on Monday, Nov. 19 during regular
class time.

You will need to know how to define the following notions:
matrix operations including multiplication, linear system of
equations, RREF of a matrix, invertible matrix and matrix inverse, span, linear independence, basis, column space, null space, linear transformation, projection, determinant,
eigenvalues and eigenvectors, discrete and continuous dynamical system, geometric growth/decay, fixed points and equilibria.

You will need to know how to perform various matrix operations, obtain RREF, solve linear systems by Gaussian elimination,
find the standard matrix of a linear transformation, use the
projection matrix formula, calculate the determinants,
find eigenvalues and eigenvectors of 2x2 matrices, 
diagonalize 2x2 matrices, find fixed points and equilibria,
sketch the phase line portrait of a differential equation.

The test will not cover MATLAB skills. You may use calculators of any sort, but
the scientific calculators will certainly suffice.

Some practice problems:

1. find the RREF of the matrix [1 2 3 4; 2 3 4 5; 3 4 5 6].
2. solve the linear system with the augmented matrix
[1 0 -1 | 3; 0 1 2 | 1; 1 -1 0 | 2];
3. find the inverse of the matrix [1 2; 3 4] or explain
why t does not exist;
4. find the eigenvalues and eigenvectors of the
matrix [1 0; 3 2];
5. find the matrix of the orthogonal projection of the
xy plane onto the line x+y=0.
6. evaluate the determinant of the matrix
[1 2 3; 2 3 1; 3 1 2].
7. find the fixed points of the discrete Beverton-Holt equation
x(n+1)= r x(n)/(K+x(n)).
8. Consider a differential equation dx/dt=x(x-a)(1-x), 0<a<1.
Find its equilibria, determine their stability, and sketch
the phase line portrait. If you pick a random initial condition
from the uniform distribution on [0,1], what is the
probability that the corresponding solution will converge
to x=0 in forward time?