Matrices Notes- Test 1 (M-maybe)
•
1.1- Systems of linear equations:
o
Row operations are reversible
o
Summary of the elimination method for 1.1:.
Check!!! Substitute into original.
o
Two fundamental Questions about a linear system:
Is the system consistent; that is, does at least one solution exist?
If a solution exists, is it the only one; that is, is the solution unique?
•
1.2- Row Reduction and Echelon Forms
o
The row reduction algorithm applies to ANY matrix, whether it’s augmented or not.
o
Theorem 1
- Uniqueness of the Reduced Echelon Form:
Each matrix is row equivalent to one and only one reduced echelon
matrix.
o
Solving a system
amounts to finding a parametric description of the solution set or determining that the solution set is empty.
o
Theorem 2
- Existence and Uniqueness Theorem:
A linear system is consistent if and only if the rightmost column of the
augmented matrix is NOT a pivot column—that is, if and only if an echelon form of the augmented matrix has NO row of the
form:
[0 … 0
b]
with b nonzero
If a linear system is consistent, then the solution set contains either (i) a unique solution, when there are
no free variables, or (ii) infinitely many solutions, when there is at least one free variable.
o
Restatement of last sentence using concept of pivot columns:
If a linear system is consistent, then the solution is unique if
and only if every column in the coefficient matrix is a pivot column; otherwise, there are infinitely many solutions.
o
Problems: 1, 5,19,21 (b),22,27M, 31M
•
1.3- Vector Equations
o
Parallelogram Rule for addition: if
u
and
v
in R
2
are represented as points in the plane, then
u
+
v
corresponds to the fourth
vertex of the parallelogram whose other vertices are
u
,
0
, and
v
.
o
If
v
1
, . . . ,
v
p
are in R
n
, then the set of all linear combinations of
v
1
, …,
v
p
is denoted by Span {
v
1
, …,
v
p
} and is called the
subset of
R
n
spanned
(or
generated
)
by
v
1
, …,
v
p
.
That is, Span {
v
1
, …,
v
p
} is the collection of all vectors that can be written
in the form
c
1
v
1
+
c
2
v
2
+ … +
c
p
v
p
with
c
1
, … ,
c
p
scalars.
So, asking whether a vector
b
is in Span {
v
1
, …,
v
p
} amounts to asking whether the vector equation
x
1
v
1
+
x
2
v
2
+ …
+
x
p
v
p
=
b
has a solution, or equivalently, asking whether the linear system with augmented matrix [
v
1
. . .
v