Linear Algebra. Math-4100, Fall 2007
Assignment 1
Due Thursday, September 13, by 4pm. (Either in class, or my mailbox in AE 301, or
under my door AE 405).
Reading
Please keep in mind that reading mathematical texts requires a serious, active work with
paper and pen to make sure that you understand what is going on. Reading for the nearest
future is in italics.
Strang, Read Sections 1.1–1.2, 2.1–2.5, 2.6–2.7.
Problems to be handed in
You are welcome to consult the text and notes and discuss the problems with other people.
However, the solutions should be yours. Please indicated on your papers, who you discussed
the problems with. Please submit extra credit problems on a separate sheet of paper.
The first four problems are slightly changed versions of problems from Strang’s book. For
an extra point a piece figure out which ones.
1. Prove that if the vectors v1, ..., vnspan space V, then so do the vectors
v1−v2,v2−v3, ..., vn−1−vn,vn
In other words, given (any) v=x1v1+... +xnvn,find y1,y2, ..., ynso that
v=y1(v1−v2) + ... +yn−1(vn−1−vn) + ynvn
2. Let Vbe a subspace of R4defined by
V={(x1,x2, x3,x4)∈R4:x1=3x2}
Find a basis of V.
3. In the space of polynomials of degree 5 consider the following two bases:
p0= 1,p1=x, p2=x2,p3=x3,p4=x4,p5=x5
p0
0= 1,p0
1=x−1,p0
2= (x−1)2, ..., p0
5= (x−1)5
(a) Find the matrix Aof transformation from {p0,p1, ..., p5}to {p0
0,p0
1, ..., p0
5},
p0
i=a0ip0+a1ip1+... +a5ip5
(b) Similarly, find the matrix Bof transformation from {p0
0,p0
1, ..., p0
5}to {p0,p1, ..., p5}.
What is the relation between Aand B?
4. Use the Schwarz inequality |v·w| ≤ kvk kwkto justify the triangle inequality, which
says: kv+wk ≤ kvk+kwk.
