Lecture 15 slides Linear Algebra.pdf - LINEAR ALGEBRA...

LINEAR ALGEBRA
SLIDES
Lecture 15
Spring 2021
Harry W. McLaughlin
i

Introduction
This is a brief lecture; it is devoted to “How to prepare
for Exam #2”
1

Exam Reference
Roughly speaking the reference for Exam #2 includes
all of lectures following those referenced for Exam #1.
However the reference for Exam #2 does include Lec-
ture #4 (Inner Products). A list of for Exam #2 is
Lectures
9
10
11
12
13
14
4
4.1
4.2
4.3
4.4
2

Prominent Takeaways
What are they? Sample calculations?
eigenvalues/vectors
principal components
inner products
Riesz Representation Theorem
adjoints
QR -factorizations
3

Exam Format
Problem #1 (35 pts.) partial credit awarded
Problem #2 (35 pts.) partial credit awarded
Problem #3 (30 pts)
(a) 5 pts. no partial credit
(b) 5 pts. no partial credit
(c) 5 pts. no partial credit
(d) 5 pts. no partial credit
(e) 5 pts. no partial credit
(f) 5 pts. no partial credit
(g) 5 pts. no partial credit
(h) 5 pts. no partial credit
4

Advanced Preparation
One of the first two problems on Exam #2 will read:
Problem
Let P
2
denote the three dimensional vector space con-
sisting of all quadratic polynomials over R endowed
with the inner product h· , ·i defined by
h p, q i =
Z
1
0
p ( x ) q ( x ) dx, ∀ p, q ∈ P
2
.
Using the Gram-Schmidt process on the basis, 1 , x, x
2