SPANNING SETS
if S = cfw_v1, v2, . . . , vk is a set of vectors in R n , then the set of all linear combinations of v1, v2, . . . , vk is
called the Span of v1, v2, . . . , vk.
It is denoted by span(v
Test 2 Review MATH 349
This is a general list of topics that you should use while studying for Test 2. This test will
cover the sections below. You should use this list as a guide while you study the
6.3
Change of Basis
Definition:
If we change the basis for a vector space V from an old basis B = cfw_u1 , u2 , . . . , un
to a new basis C = cfw_v1 , v2 , . . . , vn then for each v in V ,
(v)B = P
6.1
Vector Spaces and Subspaces
Definition: Vector Space Axioms
Let V be an arbitrary nonempty set of objects on which two operations are
defined: addition, and multiplication by scalars. By addition
5.2
Orthogonal Complements and Orthogonal Projections
Definition:
Let W be a subspace of Rn . We say that a vector V in Rn is Orthogonal to W
if v is orthogonal to every vector in W . The set of all v
3.3
The Inverse of a Matrix
Definition: Inverse
If A is a square matrix, and if a matrix B of the same size can be found such
that AB = BA = I, then A is said to be invertible (or nonsingular) and B i
3.2
Matrix Operations
Properties of Matrix Arithmetic:
1. A + B = B + A
(commutative addition)
2. A + (B + C) = (A + B) + C
3. A(BC) = (AB)C
(associative addition)
(associative multiplication)
4. A(B
3.1
Matrix Operations
Definitions:
A Matrix is a rectangular array of numbers called Entries, or Elements, of
the matrix.
A general m n matrix has the form:
a11 a12
a
21 a22
.
.
.
.
am1 am2
a1n
6.5
The Kernel and Range of a Linear Transformations
Definitions:
Let T : V W be a linear transformation.
The Kernel of T , denoted ker(T ), is the set of all vectors in V that are mapped
by T to 0 i
6.4
Linear Transformations
Definition:
If T : V W is a function from a vector space V to a vector space W , then T
is called a linear transformation from V to W if the following hold for all u and v
i
3.4
The LU Factorization
Definition:
Let A be a square matrix. A factorization of A as A = LU , where L is unit
lower triangular and U is upper triangular is called an LU Factorization of A.
A Unit Lo
5.3
The Gram-Schmidt Process and the QR Factorization
The Gram-Schmidt Process: Orthogonalization of a basis
To convert any basis cfw_x1 , x2 , . . . , xk into an orthogonal basis cfw_v1 , v2 , . . .
Test 3 Review MATH 349
This is a general list of topics that you should use while studying. This test will cover the
sections below. You should use this list as a guide while you study the suggested H
3.5
Subspaces, Basis, Dimension, and Rank
Definition:
A subset W of a vector space V is called a subspace of V if W itself is a vector
space under the addition and scalar multiplication defined on V .
DETERMINANTS
If A is an nn triangular matrix then the determinant of A is the product of the main diagonal entries:
det(A) = a11 a22 a33 ann
Let A be a square matrix. If A has a row of zeros or a colu
THE MATRIX OF LINEAR TRANSFORMATION
Let V and W be two finite-dimensional vector spaces with bases B and C, respectively, where B = cfw_v1, v2, .
. . , vn. If T : V W is a linear transformation, then
Change of Basis
If we change the basis for a vector space V from an old basis B = cfw_u1, u2, . . . , un to a new basis C = cfw_v1,
v2, . . . , vn then for each v in V , (v)B = P(v)C where the columns
LINEAR TRANSFORMATIONS
If T : V W is a function from a vector space V to a vector space W, then T is called a linear
transformation from V to W if the following hold for all u and v in V and all scala
SCALAR MULTIPLICATION
Scalar Multiplication: Multiplying a vector v by a scalar k, kv, will stretch or shrink the vector. The
direction will not change unless k < 0 in which case it will point in the
INTRO
The Geometry and Algebra of Vectors Definitions:
1.1 Vectors are line segments with specified direction and length (or magnitude).
1.2 Equivalent or Equal vectors have the same magnitude and dir
ORTHOGONAL COMPLEMENTS
Let W be a subspace of R n . We say that a vector V in R n is Orthogonal to W if v is orthogonal to every
vector in W.
The set of all vectors that are orthogonal to W is called
LENGTH AND ANGLE
Let u, v, and w be vectors in R n and let c be a scalar. Then (a) u v = v u (b) u (v + w) = u v + u w (c)
(cu) v) = c(u v) (d) u u 0 and u u = 0 if and only if u = 0
The Length or Nor
LENGTH AND ANGLE
The Triangle Inequality For all vectors u and v in R n , ku + vk kuk + kvk
The Distance between vectors u and v in R n is defined by d(u, v) = ku vk
For nonzero vectors u and v in R n
Chapter 6
Vector Spaces
6.1
Vector Spaces and Subspaces
x
. V is the set of all vectors in R2 whose first and second components are the same.
x
We verify all ten axioms of a vector space:
x+y
y
Chapter 4
Eigenvalues and Eigenvectors
4.1
Introduction to Eigenvalues and Eigenvectors
1. Following example 4.1, to see that v is an eigenvector of A, we show that Av is a multiple of v:
1
3
01
Test 4 Review MATH 349
This is a general list of topics that you should use while studying. This test will cover the
sections below. You should use this list as a guide while you study the suggested H
A linear transformation T : V W is called One-to-one if T maps distinct vectors in V to distinct vectors
in W.
T : V W is one-to-one if, for all u and v in V u 6= v implies that T(u) 6= T(v) T : V W i
A subset W of a vector space V is called a subspace of V if W itself is a vector space under the addition
and scalar multiplication defined on V
. a) If u and v are vectors in W, then u + v is in W.
b