REVIEW SHEET FOR THE
MIDTERM
Midterm
will cover the material we saw in class and studied in homework, section 5.1
through 5.6 and 6.1. (Review definitions and theorems covered in class, as well
as those highlighted in the book.) Below is the list of most fundamental
concepts that we studied as well as questions related to them.
This is NOT
a substitution for reviewing your notes, homework and quizzes that you took,
but rather a complementary hand-out to help you organize the material that we
covered.
Definitions:
- eigenvalues and eigenvectors;
- eigenspace;
- characteristic
polynomial;
- similar matrices
- diagonal
matrix
- rotation
matrix
- rotation/dilation
matrix
- inner
product
- length of
a vector
Concepts:
- basis, coordinates relative to a given
basis
- matrix of a linear transformation relative to a basis
- discrete dynamical system
- attractor,
repeller, saddle point
- trajectory of a dynamical system
Theorems:
- Theorem 1
- Theorem 2 (with proof – we did it in
class)
- Theorem 4
- Theorem 5: “Diagonalization
theorem”
- Theorem 6
- Theorem 8
- Theorem 9
Skills:
you should be able to
-
Compute characteristic polynomials, eigenvalues and eigenvectors
-
Make a decision of whether matrix is diagonalizable. Diagonalize
- Compute
powers of matrices using diagonalization
- Compute
matrices of linear transformations relative to given bases
- Recognize
rotation and rotation/dilation matrices
- For
a matrix with complex eigenvalues take it to the form PCP-1 with $C$
a rotation/dilation matrix; apply this concept to computing powers of matrices.
- Write
general formula for a state xk of a discrete dynamical system
-
Classify the origin as an attractor, repellor or a saddle point of a dynamical
system
- Sketch
typical trajectories of a dynamical system. Determine directions of greatest
attraction/repulsion
- Compute
inner product, length, distance, angle between two vectors
- Compute
unit vector in a given direction
- Determine
whether vectors are orthogonal