MAT 531 Homework 2
Spring 2014
1. Let V be a vector space over R. Given vectors u, v, w V , prove
that cfw_u, v, w is linearly independent if and only if cfw_u, u + v, u + v + w
is linearly independent.
2. Determine whether cfw_1 + x + 4x2, 3 4x 5x2, 2 5x
Eigenvalues and eigenvectors
Denition. Let A Mn,n (F). A scalar F is
called an eigenvalue of the matrix A if Av = v
for a nonzero column vector v Fn .
The vector v is called an eigenvector of A
belonging to (or associated with) the eigenvalue .
Remarks. A
Determinants : Properties
Michael Freeze
MAT 531: Linear Algebra
UNC Wilmington
Spring 2014
1 / 10
An Oracle for Singularity
Problem
We would like to dene a function of the entries of a square
matrix A whose vanishing characterizes the singularity of A.
S
Basis and coordinates
If cfw_v1 , v2 , . . . , vn is a basis for a vector space V ,
then any vector v V has a unique representation
v = x1 v1 + x2 v2 + + xn vn ,
where xi F. The coecients x1 , x2 , . . . , xn are
called the coordinates of v with respect
Basis
Denition. Let V be a vector space. A linearly
independent spanning set for V is called a basis.
Theorem A nonempty set S V is a basis for V
if and only if any vector v V is uniquely
represented as a linear combination
v = r1 v1 + r2 v2 + + rk vk , w
Linear mapping = linear transformation
Denition. Given vector spaces V1 and V2 , a
mapping L : V1 V2 is linear (or F-linear) if
L(x + y) = L(x) + L(y),
L(r x) = rL(x)
for any x, y V1 and r F.
A linear mapping : V F is called a linear
functional on V .
If
Span
Let V be a vector space over a eld F and let S be
a subset of V .
Denition. The span of the set S, denoted
Span(S), is the smallest subspace W V that
contains S.
Theorem If S is not empty then Span(S) consists
of all linear combinations r1 v1 + r2 v2
Vector space
A vector space is a set V equipped with two operations,
addition V V (x, y) x + y V and scalar
multiplication R V (r , x) r x V , that have the
following properties:
VS1. x + y = y + x for all x, y V .
VS2. (x + y) + z = x + (y + z) for all x
Classical vectors
Vector is a mathematical concept characterized by
its magnitude and direction.
Scalar is a mathematical concept characterized by
its magnitude and, possibly, sign.
Scalar is a real number (positive or negative).
Many physical quantities
Basis and coordinates
If cfw_v1 , v2, . . . , vn is a basis for a vector space V ,
then any vector v V has a unique representation
v = x 1 v1 + x 2 v2 + + x n vn ,
where xi F. The coecients x1, x2, . . . , xn are
called the coordinates of v with respect
Let V be a vector space and = [v1 , . . . , vn ] be an
ordered basis for V .
Theorem 1 The coordinate mapping C : V Fn
given by C (v) = [v] is linear and invertible (i.e.,
one-to-one and onto).
Let W be another vector space and
= [w1 , . . . , wm ] be an
Matrix of a linear transformation (revisited)
Let V , W be vector spaces and f : V W be a linear map.
Let = [v1 , v2 , . . . , vn ] be a basis for V and g1 : V Fn
be the coordinate mapping corresponding to this basis.
Let = [w1 , . . . , wm ] be a basis f
Transpose of a matrix
Denition. Given a matrix A, the transpose of A,
denoted At , is the matrix whose rows are columns
of A (and whose columns are rows of A). That is,
if A = (aij ) then At = (bij ), where bij = aji .
1 4
t
1 2 3
Examples.
= 2 5 ,
4 5 6
MAT 531
Midterm Review Guide
The midterm will cover material from Chapters 1 through 3 of the text and material discussed in
class or studied in homework exercises. Some of the most important skills with which you should
have facility include the followin
Elementary row operations for matrices:
(1) to interchange two rows;
(2) to multiply a row by a nonzero scalar;
(3) to add the i th row multiplied by some scalar r
to the jth row.
Remark. Rows are added and multiplied by scalars
as vectors (namely, row ve
Orthogonality
Let V be an inner product space with an inner
product , .
Denition 1. Vectors x, y V are said to be
orthogonal (denoted x y) if x, y = 0.
Denition 2. A vector x V is said to be
orthogonal to a nonempty set Y V (denoted
x Y ) if x, y = 0 for
Diagonalization of Linear Operators on
Inner Product Spaces
Michael Freeze
MAT 531: Linear Algebra
UNC Wilmington
Spring 2014
1 / 27
Problem
Let T : V V be a linear operator on an
inner product space V .
What conditions guarantee that V has an
orthonormal
The Adjoint of a Linear Operator
Michael Freeze
MAT 531: Linear Algebra
UNC Wilmington
Spring 2014
1 / 18
Adjoint Operator
Let L : V V be a linear operator on an inner product space
V.
Denition
The adjoint of L is a transformation L : V V satisfying
L(x),
MAT 531 Homework 1
Spring 2014
1. Determine any real numbers c for which
W = cfw_(a1, a2, a3) R3 : a1 4a2 a3 = c
is a subspace of R3.
2. Let V be a vector space over a eld F . Prove that a subset W of V
is a subspace of V if and only if 0 W and ax + y W w
MAT 531 Homework 3
Spring 2014
1. Let and be the standard ordered bases for R3 and R2 respectively. Compute [T ] for the linear transformation T : R3 R2
dened by T (a1, a2, a3) = (5a1 a2 + 4a3, a2 + a3).
2. Dene T : M22(R) P2(R) by T
Let =
a b
c d
1 0
0 1
MAT 531 Homework 4
Spring 2014
1. Apply cofactor
expansion along the third row to evaluate the deter
1 0 2
minant of A = 0 1 5 .
1 3 0
2. Prove that det(kA) = k n det(A) for any A Mnn(F ).
3. A matrix Q Mnn(R) is called orthogonal if QQt = I. Prove
that
MAT 531 Homework 6
Spring 2014
1. Let V = Mmn(R) and A V . Prove that |A| = max |Aij | denes
i,j
a norm on V .
2. Let be a basis for a nite-dimensional inner product space V and
let x, y V . Prove that if x, z = y, z for all z , then x = y.
3. Let T be a
MAT 531 Homework 7
Spring 2014
1. Let V be an inner product space, and let W be a nite-dimensional
subspace of V . If x W , prove that there exists y V such that
y W but x, y = 0.
2. Let V and W be subspaces of a nite-dimensional inner product
space. Prov
MAT 531 Homework 5
Spring 2014
1. Let T : R3 R3 be the linear transformation dened by
T (a, b, c) = (2a + 5b + 3c, 5a + 2b + c, 8c).
Find the eigenvalues of T and an ordered basis for R3 such that
[T ] is a diagonal matrix.
1 5 0
2. Show that A = 5 1 0 is
Jordan Canonical Form
Michael Freeze
MAT 531: Linear Algebra
UNC Wilmington
Spring 2014
1 / 16
Jordan Blocks
Denition
A Jordan block is an n n matrix of
1 0
0 1
0 0
J = . . .
. . .
. . .
0 0 0
0 0 0
the form
0 0
0 0
0 0
. .
. .
. .
1
0
Examples
1 0
Rational Canonical Form
Michael Freeze
MAT 531: Linear Algebra
UNC Wilmington
Spring 2014
1/6
What to do when the characteristic polynomial
does not split
For some linear operators T on V , the characteristic
polynomial does not split. Nevertheless, there
Generalized Eigenvectors
Michael Freeze
MAT 531: Linear Algebra
UNC Wilmington
Spring 2014
1 / 19
What to do when Diagonalization is Not Possible
Not every linear operator is diagonalizable, but we can show
that for any linear operator T on V whose charac
Positive Semidenite Operators
Michael Freeze
MAT 531: Linear Algebra
UNC Wilmington
Spring 2014
1/9
Gramian Matrices
Denition
An n n real matrix A is said to be a Gramian matrix when
there exists a real (square) matrix B such that A = B T B.
Proposition
L
Euclidean structure
In addition to the linear structure (addition and
scaling), space R3 carries the Euclidean structure:
length of a vector: |x|,
angle between vectors: ,
dot product: x y = |x| |y| cos .
C
y
A
x
B
Euclidean structure
Properties of vec
Diagonalizability
Michael Freeze
MAT 531: Linear Algebra
UNC Wilmington
Spring 2014
1 / 15
Diagonalizable Operators and Matrices
Denition
A linear operator T on a nite-dimensional vector space V is
called diagonalizable if there is an ordered basis for V