Math 171B: Mathematical Programming
Instructor: Michael Holst
Spring Quarter 2005
Homework Assignment #1
Due Friday April 1, 2005
Our goal in this homework is to review some basic concepts from linear algebra and from calculus of real and mul-
tivariate functions. In particular, from linear algebra we will need matrices, eigenvalues and eigenvectors, singular
values, and related concepts. From calculus, we will need to recall the definitions of continuity and differentiation, in
the cases of both real-valued and vector-valued functions of one and many variables. Key tools throughout the course
will be Taylor series and the Taylor Remainder Theorem. We will do a couple of simple implementation problems to
start us off using Matlab. This homework covers mainly material from Chapter 1 of the textbook, and is mainly a
review of things you (should) already know.
The starred exercises are those that require the use of Matlab. You MUST do the Matlab problems to get
credit for the homework.
Exercise 1.1. If xis an eigenvector of A, show that βx is also an eigenvector for any β6= 0. What is the associated
eigenvalue? Use this result to show that the unit vector x/kxkformed from an eigenvector xis also an eigenvector
of Acorresponding to the same eigenvalue as that of x.
Exercise 1.2. Let (x, y) : V7→ Rbe an inner-product on a vector space Vwith associated scalar field R. We know
that (x, y) must satisfy the three properties of an inner-product:
1. (x, x)≥0,∀x∈V, (x, x) = 0 iff x= 0.
2. (x, y) = (y, x),∀x, y ∈V.
3. (αx +βy, z ) = α(x, z) + β(y, z),∀α, β ∈R,∀x, y, z ∈V.
Use these three properties to show that the induced norm kxk= (x, x)1/2satisfies the three properties of a norm:
1. kαxk=|α| kxk,∀α∈R,∀x∈V.
2. kxk ≥ 0,∀x∈V, kxk= 0 iff x= 0.
3. kx+yk ≤ kxk+kyk,∀x, y ∈V.
Hint: Showing the first two properties is very easy; to show the last property (triangle inequality), assume the
Cauchy-Schwarz inequality holds: |(x, y)| ≤ kxk kyk.
Exercise 1.3. Let F(x) denote a twice-differentiable function of one variable. Assuming only the mean-value theo-
rem of integral calculus: F(b) = F(a) + Rb
aF0(t)dt, derive the following variants of the Taylor-series expansion with
integral remainder:
(a) F(x+h) = F(x) + Rx+h
xF0(t)dt.
(b) F(x+h) = F(x) + hR1
0F0(x+ξh)dξ.
(c) F(x+h) = F(x) + hF 0(x) + hR1
0[F0(x+ξh)−F0(x)] dξ.
(d) F(x+h) = F(x) + hF 0(x) + h2R1
0F00(x+ξh)(1 −ξ)dξ . (Hint: Try expanding F0(x+h) using a formula like
part (b) and then differentiate with respect to husing the chain rule.)
