LECTURE 1 I. Inverse matrices
We return now to the problem of solving linear equations. Recall that
we are trying to nd x such that
Ax = y.
Recall: there is a matrix I such that
Ix = x
for all x Rn . It follows that
IA = A
for all n n matrices A.
I. Column space, nullspace, solutions to the Basic Problem
Let A be a m n matrix, and y a vector in Rm . Recall the fact:
Theorem: Ax = y has a solution if and only if y is in the column space
R(A) of A.
Now lets prove a new theorem which talks
Start with Strangs elegant description of errors and blunders. Errors are
unavoidable aspects of any computation, whether because of a computers
inherent limitations or our own human capacity for mistakes. Blunders are
avoidable mistakes that c
0. Distinct eigenvalues
I havent gotten around to stating the following important theorem:
Theorem: A matrix with n distinct eigenvalues is diagonalizable.
Proof (Sketch) Suppose n = 2, and let 1 and 2 be the eigenvalues,
v1 , v2 the e
Second mid-term exam for Math 204
June 22, 2000
Rules and regulations
This exam is due at noon on Wednesday, April 12.
The exam is open-book; we ask you not to consult books other than your textbook,
or other people. We also ask that you do not use comp
First mid-term exam for Math 204
The exam consists of three questions. You are expected to work on the exam in collaboration
with the other members of your group. The task of writing up the problems, however, should
be divided among the group, with each