Linear Algebra
(Math 301)
Professor Donna Calhoun
Office : MG 241A
Office Hours : Tuesday 11-12
Spring 2013
http://math.boisestate.edu/~calhoun/teaching/Math301_Spring2013
Thursday, January 24, 13
Textbook
Introduction to Linear
Algebra, Fourth Edition
Gilbert Strang
$64 New (on Amazon)
Hardcover: 584 pages
Publisher: Wellesley Cambridge Press; 4 edition (February 10, 2009)
Language: English
ISBN-10: 0980232716
ISBN-13: 978-0980232714
Thursday, January 24, 13
Algebra?
Solve
3x 2
qua
drat
ic
for x
:
5x +
6=
x=
form
p
b±
u
l
a
2
b
4a c
2a
0
:
s
l
a
i
m
o
lyn
o
p
g
n
i
r
Facto
+ 331
3
x
7
4 + 3x
3
x
6
5x
( x 1/
Po
we
2
r
y 3/
2
)
1
(x
law
s
2
3y 5
)
Thursday, January 24, 13
1
1
x
+
1
y
=1
!
plif
y
:
x2
9
3x
+9
Partial fractions
A
Bx
+
x + 3 (x + 3)(x + 1)
Get y in terms of x :
q
Sim
y=
x
x
1
o
p
x
e
,
s
l
Radica
p
x 3 ,
p 3 27
x
s
g
o
l
,
s
nential
b)
+
a
(
n
a eb el
e
Algebra?
Linear
equations
Thursday, January 24, 13
Algebra?
Linear
equations
3x = 7
7
x = ⇡ 2.33333
3
Solution is a scalar
Thursday, January 24, 13
Algebra?
Linear
equations
3x = 7
7
x = ⇡ 2.33333
3
Solution is a scalar
2x
y=1
x+y =3
4
x = ⇡ 1.333
3
5
y = ⇡ 1.666
3
Solution is a point in the plane
Thursday, January 24, 13
3
1
3
Algebra?
Linear
equations
3x = 7
7
x = ⇡ 2.33333
3
Solution is a scalar
2x
y=1
x+y =3
4
x = ⇡ 1.333
3
5
y = ⇡ 1.666
3
Solution is a point in the plane
Thursday, January 24, 13
The equations are linear
because no powers (other
than “0” or “1”) or
products (“xy”) of x or y
appear
3
1
3
Linear equations
2x + y
4x
x = ?,
5z = 1
5y + z = 2
6x + y + 2z =
5
y = ?,
z=?
Solution is a point in three
dimensional space
Each equation describes a line in the plane, or a plane in
three-dimensional space. The solution (if one exists) is the
intersection of the two lines or three planes.
3
1
Thursday, January 24, 13
z
3
y
x
Intersection of the
three planes
What might we ask about the system?
Thursday, January 24, 13
What might we ask about the system?
2x + y
4x
5z = 1
5y + z = 2
6x + y + 2z =
Thursday, January 24, 13
x = ?,
5
y = ?,
z=?
How do we know
there is a solution?
What might we ask about the system?
2x + y
4x
5z = 1
5y + z = 2
6x + y + 2z =
In two dimensions
Parallel lines - no
solution
Thursday, January 24, 13
x = ?,
5
y = ?,
z=?
How do we know
there is a solution?
What might we ask about the system?
2x + y
4x
5z = 1
x = ?,
5y + z = 2
6x + y + 2z =
5
3
1
Thursday, January 24, 13
z=?
How do we know
there is a solution?
In two dimensions
Parallel lines - no
solution
y = ?,
3
Exactly one solution
What might we ask about the system?
2x + y
4x
5z = 1
x = ?,
5y + z = 2
6x + y + 2z =
y = ?,
z=?
How do we know
there is a solution?
5
In two dimensions
3
1
Parallel lines - no
solution
Thursday, January 24, 13
3
Exactly one solution
Co-linear - infinite
number of solutions
How do we extend this idea?
12w + 4x + 23y + 9z = 0
2u + v + 5w
5u + v
8u
2x + 2y + 8z = 1
6w + 2x + 4y
4v
5w
x
z=6
7y = 7
11u + 3v + 9x + y + 9z = 11
3u
2v
8w
15x + 5y
6z = 45
Does this system have a solution?
How do we find the solution?
Thursday, January 24, 13
Other types of systems?
Thursday, January 24, 13
Other types of systems?
3x + 8y =
4
an “underdetermined system”
(not enough equations)
Question : How do we describe
all of the solutions?
Thursday, January 24, 13
Other types of systems?
3x + 8y =
4
an “underdetermined system”
(not enough equations)
Question : How do we describe
all of the solutions?
Answer: A point (x,y) such that
for any x,
3x 1
y=
+
8
2
All of the solutions lie on a line
Thursday, January 24, 13
Other types of systems?
3x + 8y =
4
3x = 5
x=1
an “underdetermined system”
(not enough equations)
an “overdetermined system”
(too many equations)
Question : How do we describe
all of the solutions?
Question : Is there a “best”
solution?
Answer: A point (x,y) such that
for any x,
3x 1
y=
+
8
2
All of the solutions lie on a line
Thursday, January 24, 13
Other types of systems?
3x + 8y =
4
an “underdetermined system”
(not enough equations)
Question : How do we describe
all of the solutions?
Answer: A point (x,y) such that
for any x,
3x 1
y=
+
8
2
All of the solutions lie on a line
3x = 5
x=1
an “overdetermined system”
(too many equations)
Question : Is there a “best”
solution?
Answer: Find solution
“closest” to solutions to
each equation.
8
x
b=
not obvious!
5
3
Thursday, January 24, 13
5
More generally :
Thursday, January 24, 13
More generally :
u
v + 3x + 8y + 5z =
3u + 2v
x
4
y+z =0
u + 2v + x + y + 5z = 10
an “underdetermined system”
(not enough equations)
Thursday, January 24, 13
More generally :
u
v + 3x + 8y + 5z =
3u + 2v
x
4
y+z =0
u + 2v + x + y + 5z = 10
an “underdetermined system”
(not enough equations)
Question : How do we describe
all of the solutions?
Thursday, January 24, 13
More generally :
u
v + 3x + 8y + 5z =
3u + 2v
x
4
y+z =0
u + 2v + x + y + 5z = 10
an “underdetermined system”
(not enough equations)
Question : How do we describe
all of the solutions?
Answer: ???
Thursday, January 24, 13
More generally :
u
v + 3x + 8y + 5z =
3u + 2v
x
4
y+z =0
u + 2v + x + y + 5z = 10
an “underdetermined system”
(not enough equations)
Question : How do we describe
all of the solutions?
Answer: ???
Thursday, January 24, 13
2x
y=7
x + 6y = 3
2x
2y = 1
x + y = 12
x
y=3
an “overdetermined system”
(too many equations)
More generally :
u
v + 3x + 8y + 5z =
3u + 2v
x
4
y+z =0
u + 2v + x + y + 5z = 10
an “underdetermined system”
(not enough equations)
Question : How do we describe
all of the solutions?
Answer: ???
Thursday, January 24, 13
2x
y=7
x + 6y = 3
2x
2y = 1
x + y = 12
x
y=3
an “overdetermined system”
(too many equations)
Question : Is there a “best”
solution?
More generally :
u
v + 3x + 8y + 5z =
3u + 2v
x
4
y+z =0
u + 2v + x + y + 5z = 10
an “underdetermined system”
(not enough equations)
Question : How do we describe
all of the solutions?
Answer: ???
Thursday, January 24, 13
2x
y=7
x + 6y = 3
2x
2y = 1
x + y = 12
x
y=3
an “overdetermined system”
(too many equations)
Question : Is there a “best”
solution?
Answer: Find solution
“closest” to solutions to
each equation.
Where do these equations come from?
Thursday, January 24, 13
Where do these equations come from?
Suppose you have several samples of a material that is made
up of two distinct components. You (somehow) know the
volume ⌫1 , ⌫2 each component takes up in the sample, but
not their respective densities ⇢1 , ⇢2. You can find the density
of each component using the following model if you know the
weight ! of each sample
Thursday, January 24, 13
Where do these equations come from?
Suppose you have several samples of a material that is made
up of two distinct components. You (somehow) know the
volume ⌫1 , ⌫2 each component takes up in the sample, but
not their respective densities ⇢1 , ⇢2. You can find the density
of each component using the following model if you know the
weight ! of each sample
Model :
⇢1 ⌫ 1 + ⇢2 ⌫ 2 = !
Thursday, January 24, 13
Where do these equations come from?
Suppose you have several samples of a material that is made
up of two distinct components. You (somehow) know the
volume ⌫1 , ⌫2 each component takes up in the sample, but
not their respective densities ⇢1 , ⇢2. You can find the density
of each component using the following model if you know the
weight ! of each sample
Model :
⇢1 ⌫ 1 + ⇢2 ⌫ 2 = !
Data : (⌫1 , ⌫2 , !)
(4.12, 5.39, 1.09)
(4.13, 5.41, 1.20)
(3.91, 5.32, 1.11)
(3.89, 5.11, 1.02)
(2.11, 4.95, 1.05)
Thursday, January 24, 13
Where do these equations come from?
Suppose you have several samples of a material that is made
up of two distinct components. You (somehow) know the
volume ⌫1 , ⌫2 each component takes up in the sample, but
not their respective densities ⇢1 , ⇢2. You can find the density
of each component using the following model if you know the
weight ! of each sample
Model :
⇢1 ⌫ 1 + ⇢2 ⌫ 2 = !
Data : (⌫1 , ⌫2 , !)
Solve to get ⇢1 , ⇢2
(4.12, 5.39, 1.09)
4.12⇢1 + 5.39⇢2 = 1.09
(4.13, 5.41, 1.20)
4.13⇢1 + 5.41⇢2 = 1.20
(3.91, 5.32, 1.11)
3.91⇢1 + 5.32⇢2 = 1.11
(3.89, 5.11, 1.02)
3.89⇢1 + 5.11⇢2 = 1.02
(2.11, 4.95, 1.05)
4.21⇢1 + 4.95⇢2 = 1.05
More equations than unknowns
Thursday, January 24, 13
Where do these equations come from?
Suppose you have several samples of a material that is made
up of two distinct components. You (somehow) know the
volume ⌫1 , ⌫2 each component takes up in the sample, but
not their respective densities ⇢1 , ⇢2. You can find the density
of each component using the following model if you know the
weight ! of each sample
Model :
⇢1 ⌫ 1 + ⇢2 ⌫ 2 = !
“Best” solution
⇢b1 = 0.0371
⇢b2 = 0.1804
Thursday, January 24, 13
Data : (⌫1 , ⌫2 , !)
Solve to get ⇢1 , ⇢2
(4.12, 5.39, 1.09)
4.12⇢1 + 5.39⇢2 = 1.09
(4.13, 5.41, 1.20)
4.13⇢1 + 5.41⇢2 = 1.20
(3.91, 5.32, 1.11)
3.91⇢1 + 5.32⇢2 = 1.11
(3.89, 5.11, 1.02)
3.89⇢1 + 5.11⇢2 = 1.02
(2.11, 4.95, 1.05)
4.21⇢1 + 4.95⇢2 = 1.05
More equations than unknowns
Systems with millions of unknowns
•
•
•
1.23 million “degrees of
freedom” (DOF).
Solves in 6.8 minutes on
a desktop computer,
Every point on the
vehicle is an unknown.
DOF=number of equations
Thursday, January 24, 13