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, 6.2. (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

- orthogonal complement

- orthogonal set

- orthogonal basis

- orthonormal basis

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

- normalization of an orthogonal basis

Theorems:

-         Theorems 1 – 18c from class.

From the book (note that most of them are duplicate to the ones we studied in class):

Ch. V:

- Theorem 1

-     Theorem 2

-     Theorem 4

-     Theorem 5: “Diagonalization theorem”

- Theorem 6

- Theorem 7

- Theorem 8

Ch. 6

- Theorem 3

- Theorem 4

- Theorem 5

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 x_k of a discrete dynamical system

-         Classify the origin as an attractor, repeller 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 a given set is orthogonal

- Compute coordinates of a given vector relative to an orthogonal basis