lesson02 vectors and matrices slides
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Lesson 2Vectors and Matrices
Math 20
September 21, 2007
Announcements
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I Problem Set 1 is on the course web site. Due September 26.
I Office Hours: Mondays 1–2pm, Tuesdays 3–4pm, Wednesdays1–3pm (SC 323)
I Course material on website, Facebook
Vectors
There are some objects which are easily referred to collectively.
Example
The position of me on this floor can be described by two numbers.It might be
v =
(123
),
where each unit is one foot, measured from two perpendicularwalls.
Vectors
There are some objects which are easily referred to collectively.
Example
The position of me on this floor can be described by two numbers.
It might be
v =
(123
),
where each unit is one foot, measured from two perpendicularwalls.
Vectors
There are some objects which are easily referred to collectively.
Example
The position of me on this floor can be described by two numbers.It might be
v =
(123
),
where each unit is one foot, measured from two perpendicularwalls.
Example
Suppose I eat two eggs, three slices of bacon, and two slices oftoast for breakfast.
Then my breakfast can be summarized by theobject
b =
232
.
Example
Suppose I eat two eggs, three slices of bacon, and two slices oftoast for breakfast. Then my breakfast can be summarized by theobject
b =
232
.
Example
Suppose eggs cost $1.39 per dozen, bacon costs $2.49 per pound,and bread costs $1.99 per loaf. Assume a pound of bacon has 16slices, as does a loaf of bread.
Then the price per “unit” ofbreakfast is
p =
1.39/122.49/161.99/16
=
0.120.160.12
Example
Suppose eggs cost $1.39 per dozen, bacon costs $2.49 per pound,and bread costs $1.99 per loaf. Assume a pound of bacon has 16slices, as does a loaf of bread. Then the price per “unit” ofbreakfast is
p =
1.39/122.49/161.99/16
=
0.120.160.12
There is no end to the quantities that can be expressed collectivelylike this:
I stock portfolios
I (and prices)
I weather conditions
I Physical state (position, velocity)
I etc.
Matrices
In other cases numbers naturally line up into arrays. This is oftenthe case when you have two finite sets of objects and there is anumber corresponding to each pair of objects, one from each set.
Example
Pancakes, crepes, and blintzes are three types of flat breakfastconcoctions, but they have different ingredients. The ingredientscan be arranged like this:
Ingredient Pancakes Crepes Blintzes
Flour (cups) 112
12 1
Water (cups) 0 14 0
Milk (cups) 112
12 1
Eggs 2 2 3Oil (Tbsp) 3 2 2
The important information about this table is simply the numbers:
A =
1.5 0.5 10 0.25 0
1.5 0.5 12 2 33 2 2
Example
Pancakes, crepes, and blintzes are three types of flat breakfastconcoctions, but they have different ingredients. The ingredientscan be arranged like this:
Ingredient Pancakes Crepes Blintzes
Flour (cups) 112
12 1
Water (cups) 0 14 0
Milk (cups) 112
12 1
Eggs 2 2 3Oil (Tbsp) 3 2 2
The important information about this table is simply the numbers:
A =
1.5 0.5 10 0.25 0
1.5 0.5 12 2 33 2 2
Example
Here is a floorplan of my apartment:
Hall
The plan can be expressed as a graph with vertices for rooms andedges for doorways or passages between the rooms.
Hall
Kitchen
Laundry
LR
SR
BathMBR
Office BR2
Then you can make form a table of incidences:
H
K
L
LR
SR
BatMBR
O BR2
A =
S
R
LR
MB
R
Hal
l
Bat
h
Kit
Lau
nd
ry
Offi
ce
2nd
BR
0 1 0 0 0 0 0 0 0
1 0 0 1 0 0 0 1 00 0 0 1 0 0 0 0 00 1 1 0 1 1 0 0 00 0 0 1 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 00 1 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0
SRLRMBRHBatKitLOBR2
Then you can make form a table of incidences:
H
K
L
LR
SR
BatMBR
O BR2
A =
S
R
LR
MB
R
Hal
l
Bat
h
Kit
Lau
nd
ry
Offi
ce
2nd
BR
0 1 0 0 0 0 0 0 01 0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 0 00 1 1 0 1 1 0 0 00 0 0 1 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 00 1 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0
SRLRMBRHBatKitLOBR2
Then you can make form a table of incidences:
H
K
L
LR
SR
BatMBR
O BR2
A =
S
R
LR
MB
R
Hal
l
Bat
h
Kit
Lau
nd
ry
Offi
ce
2nd
BR
0 1 0 0 0 0 0 0 01 0 0 1 0 0 0 1 00 0 0 1 0 0 0 0 0
0 1 1 0 1 1 0 0 00 0 0 1 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 00 1 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0
SRLRMBRHBatKitLOBR2
Then you can make form a table of incidences:
H
K
L
LR
SR
BatMBR
O BR2
A =
S
R
LR
MB
R
Hal
l
Bat
h
Kit
Lau
nd
ry
Offi
ce
2nd
BR
0 1 0 0 0 0 0 0 01 0 0 1 0 0 0 1 00 0 0 1 0 0 0 0 00 1 1 0 1 1 0 0 0
0 0 0 1 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 00 1 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0
SRLRMBRHBatKitLOBR2
Then you can make form a table of incidences:
H
K
L
LR
SR
BatMBR
O BR2
A =
S
R
LR
MB
R
Hal
l
Bat
h
Kit
Lau
nd
ry
Offi
ce
2nd
BR
0 1 0 0 0 0 0 0 01 0 0 1 0 0 0 1 00 0 0 1 0 0 0 0 00 1 1 0 1 1 0 0 00 0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 00 1 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0
SRLRMBRHBatKitLOBR2
Then you can make form a table of incidences:
H
K
L
LR
SR
BatMBR
O BR2
A =
S
R
LR
MB
R
Hal
l
Bat
h
Kit
Lau
nd
ry
Offi
ce
2nd
BR
0 1 0 0 0 0 0 0 01 0 0 1 0 0 0 1 00 0 0 1 0 0 0 0 00 1 1 0 1 1 0 0 00 0 0 1 0 0 0 0 00 0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0 00 1 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0
SRLRMBRHBatKitLOBR2
Then you can make form a table of incidences:
H
K
L
LR
SR
BatMBR
O BR2
A =
S
R
LR
MB
R
Hal
l
Bat
h
Kit
Lau
nd
ry
Offi
ce
2nd
BR
0 1 0 0 0 0 0 0 01 0 0 1 0 0 0 1 00 0 0 1 0 0 0 0 00 1 1 0 1 1 0 0 00 0 0 1 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 0
0 1 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0
SRLRMBRHBatKitLOBR2
Then you can make form a table of incidences:
H
K
L
LR
SR
BatMBR
O BR2
A =
S
R
LR
MB
R
Hal
l
Bat
h
Kit
Lau
nd
ry
Offi
ce
2nd
BR
0 1 0 0 0 0 0 0 01 0 0 1 0 0 0 1 00 0 0 1 0 0 0 0 00 1 1 0 1 1 0 0 00 0 0 1 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 00 1 0 0 0 0 0 0 1
0 0 0 0 0 0 0 1 0
SRLRMBRHBatKitLOBR2
Then you can make form a table of incidences:
H
K
L
LR
SR
BatMBR
O BR2
A =
S
R
LR
MB
R
Hal
l
Bat
h
Kit
Lau
nd
ry
Offi
ce
2nd
BR
0 1 0 0 0 0 0 0 01 0 0 1 0 0 0 1 00 0 0 1 0 0 0 0 00 1 1 0 1 1 0 0 00 0 0 1 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 0 1 0 0 00 1 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0
SRLRMBRHBatKitLOBR2
Definition
We need some names for the things we’re working with:
DefinitionAn m× n matrix is a rectangular array of mn numbers arranged inm horizontal rows and n vertical columns.
A =
a11 a12 · · · a1j · · · a1n
a21 a22 · · · a2j · · · a2n...
.... . .
.... . .
...ai1 ai2 · · · aij · · · ain...
.... . .
.... . .
...am1 am2 · · · amj · · · amn
Definition
We need some names for the things we’re working with:
DefinitionAn m× n matrix is a rectangular array of mn numbers arranged inm horizontal rows and n vertical columns.
A =
a11 a12 · · · a1j · · · a1n
a21 a22 · · · a2j · · · a2n...
.... . .
.... . .
...ai1 ai2 · · · aij · · · ain...
.... . .
.... . .
...am1 am2 · · · amj · · · amn
Rows and Columns
DefinitionThe ith row of A is(
ai1 ai2 · · · aij · · · ain
).
The jth column of A is
a1j
a2j...
aij...
amj
Sometimes, just be succinct, we’ll write
A = (aij)m×n.
Rows and Columns
DefinitionThe ith row of A is(
ai1 ai2 · · · aij · · · ain
).
The jth column of A is
a1j
a2j...
aij...
amj
Sometimes, just be succinct, we’ll write
A = (aij)m×n.
Rows and Columns
DefinitionThe ith row of A is(
ai1 ai2 · · · aij · · · ain
).
The jth column of A is
a1j
a2j...
aij...
amj
Sometimes, just be succinct, we’ll write
A = (aij)m×n.
Dimensions
DefinitionThe dimension of a matrix A is the number of rows × (read “by”)the number of columns.
Example
The matrix in the pancakes-crepes-blintzes example is 5 × 3.
Example
The incidence matrix of my apartment is 9 × 9.
Note: Order is important!
Dimensions
DefinitionThe dimension of a matrix A is the number of rows × (read “by”)the number of columns.
Example
The matrix in the pancakes-crepes-blintzes example is
5 × 3.
Example
The incidence matrix of my apartment is 9 × 9.
Note: Order is important!
Dimensions
DefinitionThe dimension of a matrix A is the number of rows × (read “by”)the number of columns.
Example
The matrix in the pancakes-crepes-blintzes example is 5 × 3.
Example
The incidence matrix of my apartment is 9 × 9.
Note: Order is important!
Dimensions
DefinitionThe dimension of a matrix A is the number of rows × (read “by”)the number of columns.
Example
The matrix in the pancakes-crepes-blintzes example is 5 × 3.
Example
The incidence matrix of my apartment is
9 × 9.
Note: Order is important!
Dimensions
DefinitionThe dimension of a matrix A is the number of rows × (read “by”)the number of columns.
Example
The matrix in the pancakes-crepes-blintzes example is 5 × 3.
Example
The incidence matrix of my apartment is 9 × 9.
Note: Order is important!
Dimensions
DefinitionThe dimension of a matrix A is the number of rows × (read “by”)the number of columns.
Example
The matrix in the pancakes-crepes-blintzes example is 5 × 3.
Example
The incidence matrix of my apartment is 9 × 9.
Note: Order is important!
Vector
DefinitionAn n-vector (or simply vector) is an n × 1 or 1 × n matrix.
Example
We’ve seen many already. For each n there are also two zerovectors
0 =
0...0
or(0 · · · 0
).
In linear algebra we mostly work with column vectors.
Vector
DefinitionAn n-vector (or simply vector) is an n × 1 or 1 × n matrix.
Example
We’ve seen many already. For each n there are also two zerovectors
0 =
0...0
or(0 · · · 0
).
In linear algebra we mostly work with column vectors.
Vector
DefinitionAn n-vector (or simply vector) is an n × 1 or 1 × n matrix.
Example
We’ve seen many already. For each n there are also two zerovectors
0 =
0...0
or(0 · · · 0
).
In linear algebra we mostly work with column vectors.
Algebra of vectors
Example
My wife doesn’t like eggs, so her breakfast may take the form
b′ =
022
.
How can you express my wife’s and my breakfast for one day?
Answer.We just add the components each by each:2 + 0
3 + 22 + 2
=
254
.
Algebra of vectors
Example
My wife doesn’t like eggs, so her breakfast may take the form
b′ =
022
.
How can you express my wife’s and my breakfast for one day?
Answer.We just add the components each by each:2 + 0
3 + 22 + 2
=
254
.
Algebra of vectors: Adding
DefinitionThe sum of two n-vectors is the vector whose ith component isthe sum of the ith component of the first vector and ithcomponent of the second vector.
Looking above, we see my wife’s and my breakfast is measured bythe vector b + b′.
Algebra of vectors: Adding
DefinitionThe sum of two n-vectors is the vector whose ith component isthe sum of the ith component of the first vector and ithcomponent of the second vector.
Looking above, we see my wife’s and my breakfast is measured bythe vector b + b′.
Algebra of vectors
Example
Suppose I eat the same breakfast every day. What vectorrepresents my consumption over a week?
Answer.This vector is 7 · 2
7 · 37 · 2
=
142114
.
DefinitionThe scalar multiple of a vector v by number a (called a scalar) isthe vector whose ith component is a times the ith component of v.
So my weekly breakfast vector is 7b.
Algebra of vectors
Example
Suppose I eat the same breakfast every day. What vectorrepresents my consumption over a week?
Answer.This vector is 7 · 2
7 · 37 · 2
=
142114
.
DefinitionThe scalar multiple of a vector v by number a (called a scalar) isthe vector whose ith component is a times the ith component of v.
So my weekly breakfast vector is 7b.
Algebra of vectors
Example
Suppose I eat the same breakfast every day. What vectorrepresents my consumption over a week?
Answer.This vector is 7 · 2
7 · 37 · 2
=
142114
.
DefinitionThe scalar multiple of a vector v by number a (called a scalar) isthe vector whose ith component is a times the ith component of v.
So my weekly breakfast vector is 7b.
Algebra of vectors
Example
Suppose I eat the same breakfast every day. What vectorrepresents my consumption over a week?
Answer.This vector is 7 · 2
7 · 37 · 2
=
142114
.
DefinitionThe scalar multiple of a vector v by number a (called a scalar) isthe vector whose ith component is a times the ith component of v.
So my weekly breakfast vector is 7b.
Linear algebra of matrices
Matrices can be added and scaled the same way.
Example (1 23 4
)+
(1 −10 2
)=
(2 13 6
)
Example
4
(1 1−1 2
)=
(4 4−4 8
)
Linear algebra of matrices
Matrices can be added and scaled the same way.
Example (1 23 4
)+
(1 −10 2
)=
(2 13 6
)
Example
4
(1 1−1 2
)=
(4 4−4 8
)
Linear algebra of matrices
Matrices can be added and scaled the same way.
Example (1 23 4
)+
(1 −10 2
)=
(2 13 6
)
Example
4
(1 1−1 2
)=
(4 4−4 8
)
Linear algebra of matrices
Matrices can be added and scaled the same way.
Example (1 23 4
)+
(1 −10 2
)=
(2 13 6
)
Example
4
(1 1−1 2
)=
(4 4−4 8
)
Linear algebra of matrices
Matrices can be added and scaled the same way.
Example (1 23 4
)+
(1 −10 2
)=
(2 13 6
)
Example
4
(1 1−1 2
)=
(4 4−4 8
)
The plane
Given a vector
(ab
), we can consider not only the point (a, b) in
the plane, but the arrow that joins the origin to (a, b).
One reason for this arrow concept is that the addition of vectorscorresponds to a head-to-tail concatenation of vectors, ortail-to-tail by the parallelogram law.
The plane
Given a vector
(ab
), we can consider not only the point (a, b) in
the plane, but the arrow that joins the origin to (a, b).One reason for this arrow concept is that the addition of vectorscorresponds to a head-to-tail concatenation of vectors, ortail-to-tail by the parallelogram law.
Example
Let v =
(12
)and w =
(2−1
). Plot v, w, and v + w.
Solution
x
y
v
w
wv + w
v
Example
Let v =
(12
)and w =
(2−1
). Plot v, w, and v + w.
Solution
x
y
v
w
wv + w
v
Example
Let v =
(12
)and w =
(2−1
). Plot v, w, and v + w.
Solution
x
y
v
w
wv + w
v
Example
Let v =
(12
)and w =
(2−1
). Plot v, w, and v + w.
Solution
x
y
v
w
wv + w
v
Example
Let v =
(12
)and w =
(2−1
). Plot v, w, and v + w.
Solution
x
y
v
w
w
v + wv
Example
Let v =
(12
)and w =
(2−1
). Plot v, w, and v + w.
Solution
x
y
v
w
wv + w
v
Example
Let v =
(12
)and w =
(2−1
). Plot v, w, and v + w.
Solution
x
y
v
w
wv + w
v
In three dimensions we have to add a third “direction” to theCartesian plane. It’s typical to pretend it points out the paper orboard, but draw it foreshortened.
Example
Draw the vector
−121
.
Solution
y
z
x
v
In three dimensions we have to add a third “direction” to theCartesian plane. It’s typical to pretend it points out the paper orboard, but draw it foreshortened.
Example
Draw the vector
−121
.
Solution
y
z
x
v
In three dimensions we have to add a third “direction” to theCartesian plane. It’s typical to pretend it points out the paper orboard, but draw it foreshortened.
Example
Draw the vector
−121
.
Solution
y
z
x
v
In three dimensions we have to add a third “direction” to theCartesian plane. It’s typical to pretend it points out the paper orboard, but draw it foreshortened.
Example
Draw the vector
−121
.
Solution
y
z
x
v
In three dimensions we have to add a third “direction” to theCartesian plane. It’s typical to pretend it points out the paper orboard, but draw it foreshortened.
Example
Draw the vector
−121
.
Solution
y
z
x
v
Worksheet
Work in groups of 1–3.
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