Math 342 - Linear Algebra II
Notes
Note: (ELFY) means exercise left for you.
1. Vector Spaces
A vector space is a nonempty set V of objects, called vectors and denoted in boldface when
typed and with an arrow on top when handwritten, such that both (A) an
Math 342 - Linear Algebra II
Notes
Throughout these notes we assume V, W are nite dimensional inner product spaces over C.
1. Upper Triangular Representation
Proposition: Let T L(V ). There exists an orthonormal basis B = cfw_u1 , u2 , ., un of V
such th
Math 342 - Linear Algebra II
Homework 1
Due: Jan. 17th
(1) A Mnn is called symmetric if AT = A. Let V = cfw_A Mnn |A is symmetric..
(a) Show that V is a vector space (with the natural addition and scalar multiplication
for matrices).
(b) Determine a basis
Math 342 - Linear Algebra II
Notes
1. Inner Products and Norms
One knows from a basic introduction to vectors in Rn (Math 254 at OSU) that the length
of a vector x = (x1 x2 . xn )T Rn , denoted x , or |x|, is given by the formula
x2 + x2 + + x2 .
1
2
n
x=
Math 342 - Linear Algebra II
Notes
In this chapter we introduce spectral theory (eigenvalues, eigenvectors). Throughout this
chapter V is a non-zero nite-dimensional vector space over F, representing R or C.
We will use the notation L(V ) for L(V, V ), we
Math 342 - Linear Algebra II
Notes
We will go through this chapter very quickly as it is mostly 341 material.
1. Dierent Forms of Linear Systems
Recall that the linear system,
a11 x1 + a12 x2 + + a1n xn = b1 ,
a21 x1 + a22 x2 + + a2n xn , = b2 ,
am1 x1 +
Math 342 - Linear Algebra II
Notes
We will not go through chapter 3 (as it was covered in 341), but I expect you to know
how to calculate determinants via cofactor expansions and via row reduction. I also expect
you know the general theory of determinants
Math 342 - Linear Algebra II
Homework 1
Solutions
(1) A Mnn is called symmetric if AT = A. Let V = cfw_A Mnn |A is symmetric..
(a) Show that V is a vector space (with the natural addition and scalar multiplication
for matrices).
SOLUTION:
We already have
Math 342 - Linear Algebra II
Homework 2
Due: Jan. 24th
(1) Let V, W be vector spaces and T : V W be a linear transformation.
v1 , v2 , ., vk V .
Let
(a) Prove that if the vectors T v1 , T v2 , ., T vk are linearly independent then the
vectors v1 , v2 , .,
Math 342 - Linear Algebra II
Homework 4
Solutions
(1) Let V be an inner product space. Prove the parallelogram identity:
For all x, y V , x + y
2
+ xy
2
= 2( x
2
+ y 2)
SOLUTION:
Let V be an inner product space and x, y V . Then
x+y
2
+ xy
2
=
=
=
=
( x +
Math 342 - Linear Algebra II
Homework 5
Due: Feb. 28th
(1) Let V be an inner product space with an orthonormal basis B = cfw_v1 , v2 , ., vn .
(a) Prove:
n
For any x =
n
i vi and y =
i=1
n
i vi , (x, y) =
i=1
i i .
i=1
(b) Deduce from this Parsevals ident
Math 342 - Linear Algebra II
Homework 4
Due: Feb. 21st
(1) Let V be an inner product space. Prove the parallelogram identity:
For all x, y V , x + y
2
+ xy
2
= 2( x
2
+ y 2)
(2) Let V = Fn . Show that the so-called taxicab norm,
n
x
1
=
|xi |, satises the
Math 342 - Linear Algebra II
Homework 3
Solutions
(1) Compute the characteristic polynomial, eigenvalues, and eigenvectors of the following
(rotation) matrix. For what values of R are the eigenvalues and eigenvectors real?
A=
cos sin
sin cos
.
SOLUTION:
Math 342 - Linear Algebra II
Homework 2
Solutions
(1) Let V, W be vector spaces and T : V W be a linear transformation.
v1 , v2 , ., vk V .
Let
(a) Prove that if the vectors T v1 , T v2 , ., T vk are linearly independent then the
vectors v1 , v2 , ., vk a
Math 342 - Linear Algebra II
Homework 3
Due: Jan. 31st
(1) Compute the characteristic polynomial, eigenvalues, and eigenvectors of the following
(rotation) matrix. For what values of R are the eigenvalues and eigenvectors real?
A=
cos sin
sin cos
.
(2)