Two sets:VECTORS and SCALARS four operations: A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE.

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Presentation transcript:

Two sets:VECTORS and SCALARS four operations:

A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE GROUP with an identity called z z A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. 2 =

A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE GROUP with an identity called z z A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any vector : 1 = v v For any scalar s and for any two vectors: s ( ) = s s vw vw ( ()) For any two scalars a and b and for any vector: ( a + b ) = a b (()) For any scalar s and any vector: s  V V= For any two scalars a and b and for any vector: ( a  b ) = a ( b ) For any vector : 0 = z v z

A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE GROUP with an identity called z A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any vector : 1 = v v For any scalar s and for any two vectors: s ( ) = s s vw vw ( ()) For any two scalars a and b and for any vector: ( a + b ) = a b (()) For any scalar s and any vector: s  V V= For any two scalars a and b and for any vector: ( a  b ) = a ( b ) For any vector : 0 = z v z The set of points on the plane: real numbers ordinary arithmetic

A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE GROUP with an identity called z A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any scalar s and any vector: s  V V= The set of points on the plane: real numbers ordinary arithmetic 2 s s s

A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE GROUP with an identity called z A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any vector : 1 = v v For any scalar s and for any two vectors: s ( ) = s s vw vw ( ()) For any two scalars a and b and for any vector: ( a + b ) = a b (()) For any scalar s and any vector: s  V V= For any two scalars a and b and for any vector: ( a  b ) = a ( b ) For any vector : 0 = z v z The set of points on the plane: real numbers ordinary arithmetic

A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE GROUP with an identity called z A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. V= For any vector : 0 = z v z The set of points on the plane: real numbers ordinary arithmetic 0

A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE GROUP with an identity called z A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any vector : 1 = v v For any scalar s and for any two vectors: s ( ) = s s vw vw ( ()) For any two scalars a and b and for any vector: ( a + b ) = a b (()) For any scalar s and any vector: s  V V= For any two scalars a and b and for any vector: ( a  b ) = a ( b ) For any vector : 0 = z v z The set of points on the plane: real numbers ordinary arithmetic

A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE GROUP with an identity called z A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any vector : 1 = v v V= The set of points on the plane: real numbers ordinary arithmetic 1

A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE GROUP with an identity called z A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any vector : 1 = v v For any scalar s and for any two vectors: s ( ) = s s vw vw ( ()) For any two scalars a and b and for any vector: ( a + b ) = a b (()) For any scalar s and any vector: s  V V= For any two scalars a and b and for any vector: ( a  b ) = a ( b ) For any vector : 0 = z v z The set of points on the plane: real numbers ordinary arithmetic

A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE GROUP with an identity called z A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. V= For any two scalars a and b and for any vector: ( a  b ) = a ( b ) The set of points on the plane: real numbers ordinary arithmetic ( 4 × 3 )= 43 ( ) ( 12 )= 4 ( )

A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE GROUP with an identity called z A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any vector : 1 = v v For any scalar s and for any two vectors: s ( ) = s s vw vw ( ()) For any two scalars a and b and for any vector: ( a + b ) = a b (()) For any scalar s and any vector: s  V V= For any two scalars a and b and for any vector: ( a  b ) = a ( b ) For any vector : 0 = z v z The set of points on the plane: real numbers ordinary arithmetic

A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE GROUP with an identity called z A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any scalar s and for any two vectors: s ( ) = s s vw vw ( ()) V= The set of points on the plane: real numbers ordinary arithmetic 3 ( ) = 3 3 ( ())

A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE GROUP with an identity called z A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any vector : 1 = v v For any scalar s and for any two vectors: s ( ) = s s vw vw ( ()) For any two scalars a and b and for any vector: ( a + b ) = a b (()) For any scalar s and any vector: s  V V= For any two scalars a and b and for any vector: ( a  b ) = a ( b ) For any vector : 0 = z v z The set of points on the plane: real numbers ordinary arithmetic

A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE GROUP with an identity called z A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any two scalars a and b and for any vector: ( a + b ) = a b (()) V= The set of points on the plane: real numbers ordinary arithmetic ( ) = 2 3 (())

A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE GROUP with an identity called z A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any vector : 1 = v v For any scalar s and for any two vectors: s ( ) = s s vw vw ( ()) For any two scalars a and b and for any vector: ( a + b ) = a b (()) For any scalar s and any vector: s  V V= For any two scalars a and b and for any vector: ( a  b ) = a ( b ) For any vector : 0 = z v z The set of points on the plane: real numbers ordinary arithmetic

A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE GROUP with an identity called z A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any vector : 1 = v v For any scalar s and for any two vectors: s ( ) = s s vw vw ( ()) For any two scalars a and b and for any vector: ( a + b ) = a b (()) For any scalar s and any vector: s  V V= For any two scalars a and b and for any vector: ( a  b ) = a ( b ) For any vector : 0 = z v z The set of points on the plane real numbers ordinary arithmetic With INTEGER coordinates

A VECTOR SPACE consists of: A set of objects called VECTORS with operation The vectors form a COMMUTATIVE GROUP with an identity called z A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any scalar s and any vector: s  V V= The set of points on the plane real numbers ordinary arithmetic With INTEGER coordinates ½ Could you use integers rather than real numbers as your set of scalars?

VECTORS SCALARS

VECTORS SCALARS This is a commutative group

VECTORS SCALARS This is a field

The vectors form a COMMUTATIVE GROUP with an identity called C The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any vector : 1 = v v For any scalar s and for any two vectors: s ( ) = s s vw vw ( ()) For any two scalars a and b and for any vector: ( a + b ) = a b (()) For any scalar s and any vector: s  V For any two scalars a and b and for any vector: ( a  b ) = a ( b ) For any vector : 0 = z v z VECTORS SCALARS

The vectors form a COMMUTATIVE GROUP with an identity called C The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any vector : 1 = v v For any vector : 0 = z v z VECTORS SCALARS For any scalar s and any vector: s  V

The vectors form a COMMUTATIVE GROUP with an identity called C The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any two scalars a and b and for any vector: ( a  b ) = a ( b ) VECTORS SCALARS ( 2  2 ) A = 2 ( 2 A ) ( 1 ) A = 2 ( B ) A

The vectors form a COMMUTATIVE GROUP with an identity called C The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any vector : 1 = v v For any scalar s and for any two vectors: s ( ) = s s vw vw ( ()) For any two scalars a and b and for any vector: ( a + b ) = a b (()) For any scalar s and any vector: s  V For any two scalars a and b and for any vector: ( a  b ) = a ( b ) For any vector : 0 = z v z VECTORS SCALARS

The vectors form a COMMUTATIVE GROUP with an identity called C The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any scalar s and for any two vectors: s ( ) = s s vw vw ( ()) VECTORS SCALARS 2 ( A B ) = 2 A 2 B ( ()) 2 ( C ) = 2 A 2 B ( )) BA = C

The vectors form a COMMUTATIVE GROUP with an identity called C The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any two scalars a and b and for any vector: ( a + b ) = a b (()) VECTORS SCALARS ( ) = 1 2 (()) BBB ( 0 ) = (()) BB A C C

The vectors form a COMMUTATIVE GROUP with an identity called C The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any vector : 1 = v v For any scalar s and for any two vectors: s ( ) = s s vw vw ( ()) For any two scalars a and b and for any vector: ( a + b ) = a b (()) For any scalar s and any vector: s  V For any two scalars a and b and for any vector: ( a  b ) = a ( b ) For any vector : 0 = z v z VECTORS SCALARS

The vectors form a COMMUTATIVE GROUP with an identity f(x) = 0 A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any vector : 1 = v v For any scalar s and for any two vectors: s ( ) = s s vw vw ( ()) For any two scalars a and b and for any vector: ( a + b ) = a b (()) For any scalar s and any vector: s  V V= the set of all functions that are continuous on the interval [ 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 = z v z real numbers ordinary arithmetic

The vectors form a COMMUTATIVE GROUP with an identity f(x) = 0 V= the set of all functions that are continuous on the interval [ 0, 1 ] symbolized C [ 0, 1 ]

The vectors form a COMMUTATIVE GROUP with an identity A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. For any vector : 1 = v v For any scalar s and for any two vectors: s ( ) = s s vw vw ( ()) For any two scalars a and b and for any vector: ( a + b ) = a b (()) For any scalar s and any vector: s  V V= the set of all points on any line through the origin. In particular, the points on y = 2x For any two scalars a and b and for any vector: ( a  b ) = a ( b ) For any vector : 0 = z v z real numbers ordinary arithmetic

The vectors form a COMMUTATIVE GROUP with an identity A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. V= the set of all points on any line through the origin. In particular, the points on y = 2x real numbers ordinary arithmetic += closure The inverse of is.

V= the set of points on a line through the origin. V is a vector space. V is a SUBSET of another vector space R 2. V is called a SUBSPACE of R 2.

A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. The set of all points on a line that does NOT pass through the origin is NOT a vector space. Consider y = 2x + 1 real numbers ordinary arithmetic += NO closure

The vectors form a COMMUTATIVE GROUP with an identity A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. V= the set of all points on any plane through the origin. In particular, the points on 2x + 3y - z = 0 real numbers ordinary arithmetic closure: and are both points on the given plane. The sum is also a point on the plane.

The vectors form a COMMUTATIVE GROUP with an identity A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. V= the set of all points on any plane through the origin. real numbers ordinary arithmetic closure: ax + by + cz = 0 More generally, the points on any plane through the origin must satisfy an equation of the form: if then

The vectors form a COMMUTATIVE GROUP with an identity A set of objects called SCALARS with operations + and × ¾ ½ The scalars form a FIELD with identity for + called 0 identity for × called 1 The final operation called is a rule for combining a scalar with a vector to produce a vector. V= the set of all solutions to a HOMOGENIOUS system of linear equations real numbers ordinary arithmetic.