Two sets:VECTORS and SCALARS

four operations:

A VECTOR SPACE consists of:

A set of objects calledVECTORS with operation

The vectors forma COMMUTATIVEGROUP with anidentity called z

z

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

2 =

A VECTOR SPACE consists of:

A set of objects calledVECTORS with operation

The vectors forma COMMUTATIVEGROUP with anidentity called z

z

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any vector :1 =

v v

For any scalar s and for any two vectors:s ( ) = s s v w v w( () )

For any two scalars a and b and for any vector:( a + b ) = a b ( ( ))

For any scalar s andany vector: s V

V=

For any two scalars a and b and for any vector:( a b ) = a ( b )

For any vector :0 = zv z

A VECTOR SPACE consists of:

A set of objects calledVECTORS with operation

The vectors forma COMMUTATIVEGROUP with anidentity called

z

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any vector :1 =

v v

For any scalar s and for any two vectors:s ( ) = s s v w v w( () )

For any two scalars a and b and for any vector:( a + b ) = a b ( ( ))

For any scalar s andany vector: s V

V=

For any two scalars a and b and for any vector:( a b ) = a ( b )

For any vector :0 = zv z

The set of points on the plane:

.6

8

43

1

3

5

7

2

0

0etc

0

0

db

ca

d

c

b

a

real numbersordinary arith-metic

A VECTOR SPACE consists of:

A set of objects calledVECTORS with operation

The vectors forma COMMUTATIVEGROUP with anidentity called

z

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any scalar s andany vector: s V

V=The set of points on the plane:

.6

8

43

1

3

5

7

2

0

0etc

0

0

real numbersordinary arith-metic

2

10

6

5

3

b

ab

as

s

s

A VECTOR SPACE consists of:

A set of objects calledVECTORS with operation

The vectors forma COMMUTATIVEGROUP with anidentity called

z

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any vector :1 =

v v

For any scalar s and for any two vectors:s ( ) = s s v w v w( () )

For any two scalars a and b and for any vector:( a + b ) = a b ( ( ))

For any scalar s andany vector: s V

V=

For any two scalars a and b and for any vector:( a b ) = a ( b )

For any vector :0 = zv z

The set of points on the plane:

.6

8

43

1

3

5

7

2

0

0etc

0

0

real numbersordinary arith-metic

A VECTOR SPACE consists of:

A set of objects calledVECTORS with operation

The vectors forma COMMUTATIVEGROUP with anidentity called

z

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

V=

For any vector :0 = zv z

The set of points on the plane:

.6

8

43

1

3

5

7

2

0

0etc

0

0

real numbersordinary arith-metic

0

0

0

7

4

A VECTOR SPACE consists of:

A set of objects calledVECTORS with operation

The vectors forma COMMUTATIVEGROUP with anidentity called

z

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any vector :1 =

v v

For any scalar s and for any two vectors:s ( ) = s s v w v w( () )

For any two scalars a and b and for any vector:( a + b ) = a b ( ( ))

For any scalar s andany vector: s V

V=

For any two scalars a and b and for any vector:( a b ) = a ( b )

For any vector :0 = zv z

The set of points on the plane:

.6

8

43

1

3

5

7

2

0

0etc

0

0

real numbersordinary arith-metic

A VECTOR SPACE consists of:

A set of objects calledVECTORS with operation

The vectors forma COMMUTATIVEGROUP with anidentity called

z

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any vector :1 =

v v

V=The set of points on the plane:

.6

8

43

1

3

5

7

2

0

0etc

0

0

real numbersordinary arith-metic

1

7

4

7

4

A VECTOR SPACE consists of:

A set of objects calledVECTORS with operation

The vectors forma COMMUTATIVEGROUP with anidentity called

z

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any vector :1 =

v v

For any scalar s and for any two vectors:s ( ) = s s v w v w( () )

For any two scalars a and b and for any vector:( a + b ) = a b ( ( ))

For any scalar s andany vector: s V

V=

For any two scalars a and b and for any vector:( a b ) = a ( b )

For any vector :0 = zv z

The set of points on the plane:

.6

8

43

1

3

5

7

2

0

0etc

0

0

real numbersordinary arith-metic

A VECTOR SPACE consists of:

A set of objects calledVECTORS with operation

The vectors forma COMMUTATIVEGROUP with anidentity called

z

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

V=

For any two scalars a and b and for any vector:( a b ) = a ( b )

The set of points on the plane:

.6

8

43

1

3

5

7

2

0

0etc

0

0

real numbersordinary arith-metic

( 4 × 3 )

7

4= 4 3

7

4( )

( 12 )

7

4= 4

21

12( )

A VECTOR SPACE consists of:

A set of objects calledVECTORS with operation

The vectors forma COMMUTATIVEGROUP with anidentity called

z

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any vector :1 =

v v

For any scalar s and for any two vectors:s ( ) = s s v w v w( () )

For any two scalars a and b and for any vector:( a + b ) = a b ( ( ))

For any scalar s andany vector: s V

V=

For any two scalars a and b and for any vector:( a b ) = a ( b )

For any vector :0 = zv z

The set of points on the plane:

.6

8

43

1

3

5

7

2

0

0etc

0

0

real numbersordinary arith-metic

A VECTOR SPACE consists of:

A set of objects calledVECTORS with operation

The vectors forma COMMUTATIVEGROUP with anidentity called

z

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any scalar s and for any two vectors:s ( ) = s s v w v w( () )

V=The set of points on the plane:

.6

8

43

1

3

5

7

2

0

0etc

0

0

real numbersordinary arith-metic

3 ( ) = 3 3 ( () )

5

2

7

1

5

2

7

1

A VECTOR SPACE consists of:

A set of objects calledVECTORS with operation

The vectors forma COMMUTATIVEGROUP with anidentity called

z

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any vector :1 =

v v

For any scalar s and for any two vectors:s ( ) = s s v w v w( () )

For any two scalars a and b and for any vector:( a + b ) = a b ( ( ))

For any scalar s andany vector: s V

V=

For any two scalars a and b and for any vector:( a b ) = a ( b )

For any vector :0 = zv z

The set of points on the plane:

.6

8

43

1

3

5

7

2

0

0etc

0

0

real numbersordinary arith-metic

A VECTOR SPACE consists of:

A set of objects calledVECTORS with operation

The vectors forma COMMUTATIVEGROUP with anidentity called

z

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any two scalars a and b and for any vector:( a + b ) = a b ( ( ))

V=The set of points on the plane:

.6

8

43

1

3

5

7

2

0

0etc

0

0

real numbersordinary arith-metic

( 2 + 3 ) = 2 3 ( ( ))

8

5

8

5

8

5

A VECTOR SPACE consists of:

A set of objects calledVECTORS with operation

The vectors forma COMMUTATIVEGROUP with anidentity called

z

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any vector :1 =

v v

For any scalar s and for any two vectors:s ( ) = s s v w v w( () )

For any two scalars a and b and for any vector:( a + b ) = a b ( ( ))

For any scalar s andany vector: s V

V=

For any two scalars a and b and for any vector:( a b ) = a ( b )

For any vector :0 = zv z

The set of points on the plane:

.6

8

43

1

3

5

7

2

0

0etc

0

0

real numbersordinary arith-metic

A VECTOR SPACE consists of:

A set of objects calledVECTORS with operation

The vectors forma COMMUTATIVEGROUP with anidentity called

z

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any vector :1 =

v v

For any scalar s and for any two vectors:s ( ) = s s v w v w( () )

For any two scalars a and b and for any vector:( a + b ) = a b ( ( ))

For any scalar s andany vector: s V

V=

For any two scalars a and b and for any vector:( a b ) = a ( b )

For any vector :0 = zv z

The set of points on the plane

.6

8

43

1

3

5

7

2

0

0etc

0

0

real numbersordinary arith-meticWith INTEGER coordinates

A VECTOR SPACE consists of:

A set of objects calledVECTORS with operation

The vectors forma COMMUTATIVEGROUP with anidentity called

z

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any scalar s andany vector: s V

V=The set of points on the plane

.6

8

43

1

3

5

7

2

0

0etc

0

0

real numbersordinary arith-meticWith INTEGER coordinates

½ V

5

3

Could youuse integersrather thanreal numbersas your set ofscalars?

CBAC

BACB

ACBA

CBA

VECTORS

1022

0211

2100

210

SCALARS

1202

2101

0000

210

CAB

CBA

CCC

CBA

2

1

0

CBAC

BACB

ACBA

CBA

VECTORS

1022

0211

2100

210

SCALARS

1202

2101

0000

210

CAB

CBA

CCC

CBA

2

1

0

This is acommutative group

CBAC

BACB

ACBA

CBA

VECTORS

1022

0211

2100

210

SCALARS

1202

2101

0000

210

CAB

CBA

CCC

CBA

2

1

0

This is a

field

The vectors forma COMMUTATIVEGROUP with an

identity called C

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any vector :1 =

v v

For any scalar s and for any two vectors:s ( ) = s s v w v w( () )

For any two scalars a and b and for any vector:( a + b ) = a b ( ( ))

For any scalar s andany vector: s V

For any two scalars a and b and for any vector:( a b ) = a ( b )

For any vector :0 = zv z

CBAC

BACB

ACBA

CBAVECTORS SCALARS

1022

0211

2100

210

1202

2101

0000

210

CAB

CBA

CCC

CBA

2

1

0

The vectors forma COMMUTATIVEGROUP with an

identity called C

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any vector :1 =

v v

For any vector :0 = zv z

CBAC

BACB

ACBA

CBAVECTORS SCALARS

1022

0211

2100

210

1202

2101

0000

210

CAB

CBA

CCC

CBA

2

1

0

For any scalar s andany vector: s V

The vectors forma COMMUTATIVEGROUP with an

identity called C

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any two scalars a and b and for any vector:( a b ) = a ( b )

CBAC

BACB

ACBA

CBAVECTORS SCALARS

1022

0211

2100

210

1202

2101

0000

210

CAB

CBA

CCC

CBA

2

1

0

( 2 2 ) A = 2 ( 2 A )

( 1 ) A = 2 ( B )

A A

The vectors forma COMMUTATIVEGROUP with an

identity called C

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any vector :1 =

v v

For any scalar s and for any two vectors:s ( ) = s s v w v w( () )

For any two scalars a and b and for any vector:( a + b ) = a b ( ( ))

For any scalar s andany vector: s V

For any two scalars a and b and for any vector:( a b ) = a ( b )

For any vector :0 = zv z

CBAC

BACB

ACBA

CBAVECTORS SCALARS

1022

0211

2100

210

1202

2101

0000

210

CAB

CBA

CCC

CBA

2

1

0

The vectors forma COMMUTATIVEGROUP with an

identity called C

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any scalar s and for any two vectors:s ( ) = s s v w v w( () )

CBAC

BACB

ACBA

CBAVECTORS SCALARS

1022

0211

2100

210

1202

2101

0000

210

CAB

CBA

CCC

CBA

2

1

0

2 ( A B ) = 2 A 2 B ( () )

2 ( C ) = 2 A 2 B ( ) )B A = C

The vectors forma COMMUTATIVEGROUP with an

identity called C

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any two scalars a and b and for any vector:( a + b ) = a b ( ( ))

CBAC

BACB

ACBA

CBAVECTORS SCALARS

1022

0211

2100

210

1202

2101

0000

210

CAB

CBA

CCC

CBA

2

1

0

( 1 + 2 ) = 1 2 ( ( ))B B B

( 0 ) = ( ( ))B B A

C C

The vectors forma COMMUTATIVEGROUP with an

identity called C

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any vector :1 =

v v

For any scalar s and for any two vectors:s ( ) = s s v w v w( () )

For any two scalars a and b and for any vector:( a + b ) = a b ( ( ))

For any scalar s andany vector: s V

For any two scalars a and b and for any vector:( a b ) = a ( b )

For any vector :0 = zv z

CBAC

BACB

ACBA

CBAVECTORS SCALARS

1022

0211

2100

210

1202

2101

0000

210

CAB

CBA

CCC

CBA

2

1

0

The vectors forma COMMUTATIVEGROUP with anidentity f(x) = 0

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any vector :1 =

v v

For any scalar s and for any two vectors:s ( ) = s s v w v w( () )

For any two scalars a and b and for any vector:( a + b ) = a b ( ( ))

For any scalar s andany vector: s V

V= the set of all functionsthat are continuous on theinterval [ 0 , 1 ] symbolized C [ 0 , 1 ]

For any two scalars a and b and for any vector:( a b ) = a ( b )

For any vector :0 = zv z

real numbersordinary arith-metic

The vectors forma COMMUTATIVEGROUP with anidentity f(x) = 0

V= the set of all functionsthat are continuous on theinterval [ 0 , 1 ] symbolized C [ 0 , 1 ]

xxxf

xxxgf

xxg

xxxf

1284

735)(

73

32

2

2

2

2

The vectors forma COMMUTATIVEGROUP with anidentity

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

For any vector :1 =

v v

For any scalar s and for any two vectors:s ( ) = s s v w v w( () )

For any two scalars a and b and for any vector:( a + b ) = a b ( ( ))

For any scalar s andany vector: s V

V= the set of all points on anyline through the origin.In particular, the points ony = 2x

For any two scalars a and b and for any vector:( a b ) = a ( b )

For any vector :0 = zv z

real numbersordinary arith-metic

0

0

The vectors forma COMMUTATIVEGROUP with anidentity

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

V= the set of all points on anyline through the origin.In particular, the points ony = 2x

real numbersordinary arith-metic

0

0

6

3

6

3

2

1

2

1

4

2

4

2+ =

closure

The inverse of is .

8

4

8

4

V= the set of points on a line through the origin.

V is a vector space.

V is a SUBSET of anothervector space R2.

V is called a SUBSPACE of R2.

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

The set of all points on aline that does NOT passthrough the origin is NOTa vector space. Considery = 2x + 1

real numbersordinary arith-metic

7

3

7

3

1

1

1

1

6

2

6

2

+ =

NO closure

The vectors forma COMMUTATIVEGROUP with anidentity

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

V= the set of all points on anyplane through the origin.In particular, the points on 2x + 3y - z = 0

real numbersordinary arith-metic

0

0

0

closure:

2

0

1

3

1

0

and are both points on the given plane.

The sum is also a point on the plane.

5

1

1

The vectors forma COMMUTATIVEGROUP with anidentity

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

V= the set of all points on anyplane through the origin.

real numbersordinary arith-metic

0

0

0

closure:

ax + by + cz = 0More generally, the points on any plane through the origin must satisfyan equation of the form:

0)()()(

0

0

212121

222

111

zzcyybxxa

czbyax

czbyax if

then

The vectors forma COMMUTATIVEGROUP with anidentity

A set of objects calledSCALARS with operations + and ×

4 7 -1 ¾ 19 -67 45 ½

The scalars forma FIELD with identity for + called 0identity for × called 1

The final operation called is a rule for combining ascalar with a vector to producea vector.

V= the set of all solutions to a HOMOGENIOUSsystem of linear equations

real numbersordinary arith-metic

0

0

0

.