Math 311-101 — Final Exam Review — Summer I, 2016

General Information

Calculators. You may use scientific calculators to do numerical calculations — logs, exponentials, and so on. You may not use any calculator that has the capability of doing algebra or calculus, or of storing course material.

Other devices. You may not use cell phones, computers, or any other device capable of storing, sending, or receiving information.

Topics Covered

Change of Basis

• Coordinate vectors. Understand how to find coordinates relative to a basis. See my notes on Coordinate Vectors
• Transition matrices. Be able to find transition matrices S_E→F that change coordinates from E to F. See my notes on change of basis.

Linear Transformations

• Definition and Examples
  • Know the definition of a linear transformation. Be able to determine whether a transformation is linear.
• Matrix Representations
  • Know how to find matrix representations for linear transformations. Be able to work problems similar to ones done in class and to ones in the homework. Specific examples are matrix representations for shear transformations, rotations, dilation/contraction operations, using shear transformations to do translations via matrix multiplication, differential and integral operators.
  • If a matrix $A$ represents a linear transformation $L$ and a change of basis is made, be able to find the matrix $B$ that represents $L$ in the new basis.
  • Subspaces associated with a linear transformation: kernel (or null space), and the range, R(L) (also, L(V)). Be able to use a matrix representation for L to find these.

Inner Product Spaces

• Inner product
  • Inner product and norm. Know the definition of an inner product and its associated norm (length). Be able to show that < x,y > = y^Tx is an inner product on Rⁿ.
  • Be able to state and prove Schwarz's inequality. (See class notes for June 21, 2016.)
  • Angle and length. Be able to find the norm of a vector and to find the angle between two vectors.
• Orthogonal and orthonormal sets
  • Be able to define these terms: orthogonal and orthonormal sets; orthonormal bases.
  • Be able to verify that a set is orthogonal or orthonormal.
  • Orthogonal subspaces. Fundamental theorem of linear algebra, Fredholm alternative (notes 6/22/16), direct sum, ⊕.
  • Be able to show that sets of non-zero orthogonal vectors are linearly independent and that that vectors can be represented as in Theorem 5.5.2 (i.e., be able to PROVE this theorem).
  • Least squares. Know how to use an orthonormal set to find least squares approximations for functions.

Eigenvalue Problems

• Solving eigenvalue problems
  • Given an n×n matrix A, be able to find A's eigenvalues and their corresponding eigenvectors and eigenspaces.
• Applications. Simple electric circuits and normal modes in a spring system. (See the notes for 6/28/16).
• Diagonalization
  • Be able to diagonalize an n×n matrix A, if that is possible. The matrix A is diagonalizable and only if it is similar to a diagonal matrix D; that is, A = XDX⁻¹.
    • D = diag(λ₁, …, λ_n), where the λ's are the eigenvalues of A.
    • X = [x₁ … x_n], where x_k is the eigenvector corresponding to λ_k. The matrix X must be invertible.
    • For a diagonalizable matrix $A$, be able to find the matrix exponential $e^A$.
  • Non diagonalizable matrices. There are matrices that can't be diagonalized. These are called defective. Be able to determine whether A is diagonalizable or defective.

Vector Calculus

• Be able to compute the gradient, divergence, and curl.
• Be able to compute line integrals directly or via Green's theorem.
• Parametrized surfaces
  • Given a parametrization for a surface, be able to find these quantities: standard normal N, unit normal n, vector area element dS, and the scalar area element dS.
  • Know parametrizations for spheres, cylinders, and planes. For each of these, know N, n, dS, dS.
• Surface integrals
  • Be able to compute both scalar and vector surface integrals, either directly (Colley, section 11.2), or via Stokes's Theorem or Gauss's Theorem (Colley, section 11.3).

Practice tests

• Practice test, Fall 2003.
• Practice test, Spring 1998.
• Practice test, Summer 1991.

Updated: 6/28/2016 (fjn)