Math 311-101 — Test 2 Review — Summer I, 2012

General Information

Test 2 (Tuesday, June 26) will have 5 to 7 questions, some with multiple parts. It will cover sections 4.1-4.3, 5.4, 5.5 (skip bottom of p. 273 to top of p. 279), 6.1, (skip 6.2), 6.3 in Leon's Linear Algebra (Part I), plus the notes on change of basis and least-squares problems. Please bring an 8½×11 bluebook. Problems will be similar to ones done for homework. I will have extra office hours on Monday afternoon, 11:45 am-4 pm, and Tuesday morning, 8:30-9:30 am.

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

Linear Transformations

Definition and Examples
- Know the definition of a linear transformation. Be able to determine whether a transformation is linear.
- Simple properties. (p. 193)
- Subspaces associated with a linear transformation: kernel (or null space), image of a subspace S, L(S), and the range, L(V). Be able to find these space for examples similar to ones in the text. The best way to get these is to use a matrix representaion for L. (See below.)
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. (We will only deal with L:V → V.)
Similarity
- Change of basis. Be able to find transition matrices S_E→F that change coordinates from E to F. (See my notes on change of basis.)
- Change of basis for a linear transformation. Be able to find the matrix B of a linear transformation, relative to a basis F, given that you know its matrix A relative to a basis E. (See my notes on change of basis.) Know the definition of similar matrices.

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 work with inner products of the form < f,g > = ∫_a^b f(x)g(x)dx.
- Be able to state Schwarz's inequality and the triangle inequality.
- 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.
- 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 sqaures approximations for functions. (See my notes on least-squares problems.)

Eigenvalue Problems

Solving eigenvalue problems for an n×n matrix A
- Definition. A scalar λ is an eigenvalue of A if and only if there is a non-zero vector x for which Ax = λx. The vector x is an eigenvector associated with λ. Be able to find the eigenvalues and eigenvectors for a matrix, including cases where the eigenvalues are complex numbers.
- Characteristic polynomial.
  - p_A(λ) = det(A-λI). Know that p_A is a polynomial of degree n and that
    p_A(λ) = (-1)ⁿ(λⁿ - trace(A)λ + ... (-1)ⁿ det(A)),
    where trace(A) = a_1,1 + ... + a_n,n.
  - Know that eigenvalues of A are the same as the roots of p_A.
  - If A and B are similar, then p_B(λ) = p_A(λ). Thus, similar matrices have the same eigenvalues.
- Eigenvectors. For an eigenvalue λ, the eigenvectors are all nonzero vectors in the null space N(A − λ I). Finding a basis for N(A − λ I) is what's required. NOTE: 0 is NEVER an eigenvector.
Diagonalization
- Definition. An n× matrix A is diagonalizable and only if A is similar to a diagonal matrix D; that is, D = S⁻¹AS.
  - D = diag(λ₁, …, λ_n), where the λ's are the eigenvalues of A.
  - S = [x₁ … x_n], where x_k is the eigenvector corresponding to λ_k. The matrix S must be invertible.
- Non diagonalizable matrices. There are matrices that can't be diagonalized; for example, the shear matrix A =
  
  1 2
  
  0 1
- Be able to determine whether or not A is diagonalizable.
  - A is diagonalizable if and only if one may extract a basis for Rⁿ (or Cⁿ) from among the eigenvectors of A.
  - Know that if the eigenvalues of A are distinct, then A is diagonalizable. The converse is false, however.
- Be able to diagonalize a matrix A, if that is possible.
- Be able to compute the matrix exponential e^A.

Updated: 6/23/2012 (fjn)