August 22, 2013 Math 3312 sec 003 Fall 2013
Section 1.3: Vector Equations
Denition: A matrix that consists of one column is called a column
vector or simply a vector.
x1
with x1 and x2 any real numbers
x2
is denoted by R2 (read R two). Its the set of all
Oct. 3, 2013 Math 3312 sec 003 Fall 2013
Denition (Basis)
Denition: Let H be a subspace of a vector space V . An indexed set
of vectors B = cfw_b1 , . . . , bp in V is a basis of H provided
(i) B is linearly independent, and
(ii) H =Span(B ).
Standard Ba
Oct. 10, 2013 Math 3312 sec 003 Fall 2013
Section 4.6: Rank
Denition: The row space, denoted Row A, of an m n matrix A is
the subspace of Rm spanned by the rows of A.
Example: Express the row space of A in term of a span
2 5
8
0 17
1
3
5 1
5
A=
3 11 19 7
Oct. 8, 2013 Math 3312 sec 003 Fall 2013
Section 4.4: Coordinate Systems
Theorem: Let B = cfw_b1 , . . . , bn be a basis for a vector space V . Then
for each vector x in V , there is a unique set of scalars c1 , . . . , cn such
that
x = c1 b1 + cn bn .
D
Sept. 26, 2013 Math 3312 sec 003 Fall 2013
Section 4.1: Vector Spaces and Subspaces
Denition A vector space is a nonempty set V of objects called
vectors together with two operations called vector addition and scalar
multiplication that satisfy the follow
Sept. 24, 2013 Math 3312 sec 003 Fall 2013
Section 3.2: Properties of Determinants
Theorem: Let A be an n n matrix, and suppose the matrix B is
obtained from A by performing a single elementary row operation1 .
Then
(i) If B is obtained by adding a multip
Sept. 19, 2013 Math 3312 sec 003 Fall 2013
Section 2.3: Characterization of Invertible Matrices
Given an n n matrix A, we can think of
A matrix equation Ax = b;
A linear system that has A as its coefcient matrix;
A linear transformation T : Rn Rn dened by
Sept. 3, 2013 Math 3312 sec 003 Fall 2013
Section 1.8: Intro to Linear Transformations
Recall that the product Ax is a linear combination of the columns of
Aturns out to be a vector. If the columns of A are vectors in Rm , and
there are n of them, then
A
Sept. 5, 2013 Math 3312 sec 003 Fall 2013
Section 2.1: Matrix Operations
Recall the convenient notaton for a matrix A
a11 a12
a21 a22
A = [a1 a2 an ] = .
.
.
.
.
.
.
.
.
a1n
a2n
.
.
.
am1 am2
amn
.
Here each column is a vector aj in Rm . Well use the ad
Sept. 17, 2013 Math 3312 sec 003 Fall 2013
Section 2.2: Inverse of a Matrix
If A is an n n matrix, and there exists an n n matrix B such that
BA = AB = In ,
well say that A is nonsingular (a.k.a. invertible). And well call B the
inverse of A and write
B =
August 27, 2013 Math 3312 sec 003 Fall 2013
Vector and Matrix Equations
The vector equation
x1 a1 + x2 a2 + + xn an = b
has the same solution set as the linear system whose augmented
matrix is
[a1
a2
an
b] .
(1)
In particular, b is a linear combination of
August 29, 2013 Math 3312 sec 003 Fall 2013
Section 1.5: Solution Sets of Linear Systems
Denition A linear system
Ax = b
is said to be homogeneous if b = 0. Otherwise, it is called
nonhomogeneous.
Theorem: A homogeneous system always has at least one solu
Math 3312 sec 003 Fall 2013
We begin with a linear (algebraic) equations in n variables x1 , x2 , ., xn
for some positive integer n. A linear equation can be written in the
form
a1 x1 + a2 x2 + + an xn = b.
The numbers a1 , . . . , an are called the coefc
Aug. 20, 2013 Math 3312 sec 003 Fall 2013
Section 1.2: Row Reduction and Echelon Forms
Denition: A matrix is in echelon form (a.k.a. row echelon form) if
the following properties hold
i Any row of all zeros are at the bottom.
ii The rst nonzero number (ca
Oct. 1, 2013 Math 3312 sec 003 Fall 2013
Section 4.2: Null & Column Spaces, Linear Transformations
Denition: Let A be an m n matrix. The null space of A, denoted by
Nul A1 , is the set of all solutions of the homogeneous equation Ax = 0.
That is
NulA = cf