Advanced Topics in Numerical Analysis (Analytical Methods)
MATH GA 20113

Fall 2012
LINEAR TRANSFORMATIONS
DEFINITION:
A transformation (or function, or mapping)
T from Rn to Rm is a rule that assigns
to each vector x from Rn a vector T ()
x
in Rm. The set Rn is called the domain,
and Rm is called the codomain of T.
EXAMPLE:
Let
1 3
A =
Advanced Topics in Numerical Analysis (Analytical Methods)
MATH GA 20113

Fall 2012
ELEMENTARY ROW OPERATIONS:
1. Replace one row by the sum of itself
and a multiple of another row.
2. Interchange two rows.
3. Multiply all entries in a row by a
nonzero constant.
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
00
00
0
1
0
0
0
0
0
1
0
0
0000
0
0
0
1
0
Advanced Topics in Numerical Analysis (Analytical Methods)
MATH GA 20113

Fall 2012
(a) A 5 6 matrix has six rows. (False)
(b) Elementary row operations on an
augmented matrix never change the solution set of the associated linear system. (True)
(c) An inconsistent system has more
than one solution. (False)
Determine which matrices are i
Advanced Topics in Numerical Analysis (Analytical Methods)
MATH GA 20113

Fall 2012
SYSTEMS OF LINEAR EQUATIONS
DEFINITION 1:
A linear equation in the variables x1, . . . ,
xn is an equation that can be written in
the form
a1x1 + . . . + anxn = b,
where a1, . . . , an and b are constants,
x1, . . . , xn are variables.
EXAMPLE:
The equati
Advanced Topics in Numerical Analysis (Analytical Methods)
MATH GA 20113

Fall 2012
DEFINITION:
If A and B are m n matrices, then the
sum A + B is the m n matrix whose entries are the sums of the corresponding
entries of A and B.
EXAMPLE:
1 2 1
012
+
2 3 2
2 4 1
=
1 1 1
4 1 3
REMARK: We can add matrices only
of the same size.
EXAMPLE:
12
Advanced Topics in Numerical Analysis (Analytical Methods)
MATH GA 20113

Fall 2012
DEFINITION:
A system of linear equations is said to
be homogeneous if it can be written in
the form Ax = . Otherwise, it is non
0
homogeneous.
EXAMPLE:
3x1 + 5x2 = 0
6x1 + 2x2 = 0
HOMOGENEOUS
3x1 + 5x2 = 1
6x1 + 2x2 = 0
NONHOMOGEN.
THEOREM:
Suppose the eq
Advanced Topics in Numerical Analysis (Analytical Methods)
MATH GA 20113

Fall 2012
ECHELON FORM:
00
00 0
00 0 0
00 0 0 0000
1. All nonzero rows are above any rows
of all zeros.
2. Each leading entry of a row is in a
column to the right of the leading entry
of the row above it.
3. All entries in a column below a leading entry are
Advanced Topics in Numerical Analysis (Analytical Methods)
MATH GA 20113

Fall 2012
AREA AND VOLUME
THEOREM:
If A is a 2 2 matrix, the area of the parallelogram determined by the columns
of A is  det A. If A is a 3 3 matrix,
the volume of the parallelepiped determined by the columns of A is  det A.
CRAMERS RULE
DEFINITION:
For any n