Lesson 2Vectors and Matrices

Math 20

September 21, 2007

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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.