Week 9: 4/3-4/7
Chapter 7.3 Eigenvectors
Key points: a) Terminology: Eigenspace, eigenbasis. Note: The eigenspace
may have smaller dimension then the algebraic multiplicty
of λ .
b) Every eigenvalue has at least one eigenvector (since
det(A- λ I) = 0 and so A - λ I has a kernel.
c) Linear independence of eigenvectors
d) If S is invertible, A = SBS^-1, then A and B have the same
eigenvalues and if e_λ is one for B, then Se_λ is
the corresponding one for A.
e) The dimension of an eigenspace is no more than multiplicity
of its eigenvalue, but can be less if multiplicity is
greater than 1. Notion of geometric dimension of eigenspace
versus algebraic multiplicity.
Homework: 12, 20, 26, 36, 44, 47*, 48*
Chapter 7.4 Diagonalization
Key points: a) Diagonalization makes a linear transformation diagonal
b) Diagonalization to find powers of a matrix
c) A matrix is diagonalizable if and only if it has n
eigenvectors
Homework: 12, 16, 30, 32, 54, 38*, 58*
Chapter 7.5 Complex eigenvalues
Key points: a) Review complex numbers
b) Polar form and De Moivres’ theorem
c) Fundamental theorem of algebra and fact that square matrix
has n complex eigenvalues if counted with algebraic
multiplicity.
d) Complex eigenvalues are in conjugate pairs
Homework: 6, 24, 26, 32, 38a,b, 3*, 30*
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Solutions
Week 10: 4/10-4/14
Chapter 7.6 Stability
Key points: a) Zero is a stable equilibrium of x -> Ax if and only if all
eigenvalues have absolute value less than 1
b) Understand why this is so in terms of eigenvectors and the
flow of 'generic' point
Homework: 6, 16, 20, 28, 42, 37*
Chapter 8.1 Symmetric matrices
Key points: a) A symmetric n x n matrix has real eigenvalues, n in all.
b) Eigenvectors are orthogonal so A = SDS^-1 with S
orthogonal. Also, a symmetric matrix as a linear
transformation preserves the orthogonal complement of any
eigenspace.
Homework: 8, 10, 16, 24, 36, 20*, 30*
Chapter 9.1 Differential equations I
Key points: a) A linear dynamical system, continuous or discrete, can be
solved in closed form when the matrix is diagonalizable.
b) Phase portraits
Homework: 14, 22, 28, 40, 44, 24*, 54*
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Solutions
Week 11: 4/17-4/21
Chapter 9.2 Differential equations II
Key points: a) Derivatives of complex function of time, t -> z(t).
b) Complex exponentials and Euler’s formula
c) Solution to (∂_t)x = Ax when A has n complex eigenvectors
d) Stability if and only if real parts of all eigenvalues
are negative
e) Phase portraits
Homework: 12, 14, 22, 24, 26, 28, 38, 16 *, 21*
Otto Bretscher’s handout on nonlinear systems
Key points: a) Find the null clines
b) Find the stationary points
c) motion is vertical through x-null clines and horizontal
through y-null clines.
d) Sketch trajectories in the regions delineated by null-clines.
e) Stability of the equilibrium point is determined by the
matrix of partial derivatives: Real(λ ) < 0 for all
eigenvalues λUnderstand why this comes by using
Taylor’s theorem with remainder. Know that the borderline
case (real (λ) = 0) is usually considered to be unstable.
Homework: 1, 2, 3, 4, 5* from the handout.
Handout 10.1, Chapter 9.3; Linear differential operators
Key points: a) Spaces of functions can be viewed as vector spaces.
b) Linear differential operators can be viewed as linear
transformations.
c) Terminology: Homogeneous and inhomogeneous equations,
kernels of differential operators.
d) The notion of spanning a subset.
e) Linear independence for a set of functions.
Homework: 1-6 from Section 10.1 of the handout.
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Solutions
Week 12: 4/24-4/28
Handout 10.2, Chapter 9.3; Linear differential operators
Key points: a) Existence and ‘uniqueness’ for solutions to (∂_t^n +...)f = g
b) The explicit form of the constant coefficient case;
solutions are {e^(at): a is root of the characteristic
polynomial).
c) Know why this case is important even though special:
Taylor’s theorem (to zero’th order) allows it to locally
approximate the general case where the leading order
coefficient is non-zero.
Homework: 1-5 from Section 10.2 of the handout.
Handout 10.3 and Chapter 5.5; Fourier series
Key points: a) The norm and inner product on continous functions.
b) The notion of orthogonality. An orthonormal set of function
is linearly independent.
c) The set {1, cos(nt), sin(nt)} is orthonormal on [-π, π] and
every continous function has an expansion in this basis.
Moreover, the expansion converges pointwise to the function
if the function is differentiable. This is called the
Fourier expansion of the function.
d) Give the analogous basis for any interval.
Homework: 1-5 from Section 10.3 of the handout.
22, 24 from Chapter 5.5.
Handout 10.4; The heat equation
Key points: a) The heat equation
b) Initial conditions and boundary conditions
c) Viewing the equation as a linear operator on a vector space
of functions.
d) Solving using Fourier series
Homework: 1-6 from Section 10.4 of the handout.
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Solutions
Week 13: 5/1-5/5
Handout 10.5; Laplace’s equation
Key points: a) The Laplace equation
b) The Laplace equation as a linear transformation on a
vector space of functions.
c) Solving the Laplace equation using Fourier series.
Homework: 1-4 from Section 10.5 in the handout.
Handout 10.6; Other equations
Key points: a) The wave equation, Schroedinger’s equation.
b) Look for vector and matrix analogs to guide a search for
solutions.
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Solutions
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Math21b, Linear Algebra and Applications, Spring 2006, Clifford Taubes, Harvard University, Faculty of Arts and Sciences, Harvard University
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