1 / 37

Two sets:VECTORS and SCALARS four operations:

VECTOR SPACE. Two sets:VECTORS and SCALARS four operations:. 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 SCALARS with operations + and ×.

marnin
Download Presentation

Two sets:VECTORS and SCALARS four operations:

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. VECTOR SPACE Two sets:VECTORS and SCALARS four operations:

  2. 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 SCALARS with operations + and × A set of objects called VECTORS with operation z 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity called z The scalars form a FIELD with identity for + called 0 identity for × called 1

  3. For any two scalars aand b and for any vector: ( a b ) = a ( b ) For any scalar s and any vector: sV For any scalar sand for any two vectors: s ( ) = s s For any vector : 0 = ( ) ( ) v w v w z z v For any two scalars aand b and for any vector: ( a +b ) = a b For any vector : 1 = v v ( ) ( ) A VECTOR SPACE consists of: V= A set of objects called SCALARS with operations + and × A set of objects called VECTORS with operation z 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity called z 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.

  4. For any two scalars aand b and for any vector: ( a b ) = a ( b ) For any scalar s and any vector: sV For any scalar sand for any two vectors: s ( ) = s s For any vector : 0 = ( ) ( ) v w v w z z v For any two scalars aand b and for any vector: ( a +b ) = a b For any vector : 1 = v v ( ) ( ) EXAMPLE 1 A VECTOR SPACE consists of: V= real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × A set of objects called VECTORS with operation The set of points on the plane: z 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity called 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.

  5. s s 2 s A VECTOR SPACE consists of: V= real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × A set of objects called VECTORS with operation The set of points on the plane: z 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity called 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

  6. For any two scalars aand b and for any vector: ( a b ) = a ( b ) For any scalar s and any vector: sV For any scalar sand for any two vectors: s ( ) = s s For any vector : 0 = ( ) ( ) v w v w z z v For any two scalars aand b and for any vector: ( a +b ) = a b For any vector : 1 = v v ( ) ( ) A VECTOR SPACE consists of: V= real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × A set of objects called VECTORS with operation The set of points on the plane: z 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity called 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.

  7. For any vector : 0 = 0 z z v A VECTOR SPACE consists of: V= real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × A set of objects called VECTORS with operation The set of points on the plane: z 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity called 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.

  8. For any two scalars aand b and for any vector: ( a b ) = a ( b ) For any scalar s and any vector: sV For any scalar sand for any two vectors: s ( ) = s s For any vector : 0 = ( ) ( ) v w v w z z v For any two scalars aand b and for any vector: ( a +b ) = a b For any vector : 1 = v v ( ) ( ) A VECTOR SPACE consists of: V= real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × A set of objects called VECTORS with operation The set of points on the plane: z 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity called 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.

  9. 1 For any vector : 1 = v v A VECTOR SPACE consists of: V= real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × A set of objects called VECTORS with operation The set of points on the plane: z 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity called 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.

  10. For any two scalars aand b and for any vector: ( a b ) = a ( b ) For any scalar s and any vector: sV For any scalar sand for any two vectors: s ( ) = s s For any vector : 0 = ( ) ( ) v w v w z z v For any two scalars aand b and for any vector: ( a +b ) = a b For any vector : 1 = v v ( ) ( ) A VECTOR SPACE consists of: V= real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × A set of objects called VECTORS with operation The set of points on the plane: z 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity called 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.

  11. ( ) ( 4 × 3 ) = 4 3 ( ) ( 12 ) = 4 A VECTOR SPACE consists of: V= real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × A set of objects called VECTORS with operation The set of points on the plane: z 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity called 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 aand b and for any vector: ( a b ) = a ( b )

  12. For any two scalars aand b and for any vector: ( a b ) = a ( b ) For any scalar s and any vector: sV For any scalar sand for any two vectors: s ( ) = s s For any vector : 0 = ( ) ( ) v w v w z z v For any two scalars aand b and for any vector: ( a +b ) = a b For any vector : 1 = v v ( ) ( ) A VECTOR SPACE consists of: V= real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × A set of objects called VECTORS with operation The set of points on the plane: z 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity called 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.

  13. For any scalar sand for any two vectors: s ( ) = s s ( ) ( ) v w v w 3 ( ) = 3 3 ( ) ( ) A VECTOR SPACE consists of: V= real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × A set of objects called VECTORS with operation The set of points on the plane: z 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity called 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.

  14. For any two scalars aand b and for any vector: ( a b ) = a ( b ) For any scalar s and any vector: sV For any scalar sand for any two vectors: s ( ) = s s For any vector : 0 = ( ) ( ) v w v w z z v For any two scalars aand b and for any vector: ( a +b ) = a b For any vector : 1 = v v ( ) ( ) A VECTOR SPACE consists of: V= real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × A set of objects called VECTORS with operation The set of points on the plane: z 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity called 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.

  15. ( 2 +3 ) = 2 3 ( ) ( ) For any two scalars aand b and for any vector: ( a +b ) = a b ( ) ( ) A VECTOR SPACE consists of: V= real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × A set of objects called VECTORS with operation The set of points on the plane: z 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity called 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.

  16. For any two scalars aand b and for any vector: ( a b ) = a ( b ) For any scalar s and any vector: sV For any scalar sand for any two vectors: s ( ) = s s For any vector : 0 = ( ) ( ) v w v w z z v For any two scalars aand b and for any vector: ( a +b ) = a b For any vector : 1 = v v ( ) ( ) A VECTOR SPACE consists of: V= real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × A set of objects called VECTORS with operation The set of points on the plane: z 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity called 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.

  17. For any two scalars aand b and for any vector: ( a b ) = a ( b ) For any scalar s and any vector: sV For any scalar sand for any two vectors: s ( ) = s s For any vector : 0 = ( ) ( ) v w v w z z v For any two scalars aand b and for any vector: ( a +b ) = a b For any vector : 1 = v v ( ) ( ) EXAMPLE 2 A VECTOR SPACE consists of: V= real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × A set of objects called VECTORS with operation The set of points on the plane With INTEGER coordinates z 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity called 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.

  18. Could you use integers rather than real numbers as your set of scalars? ½ A VECTOR SPACE consists of: V= real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × A set of objects called VECTORS with operation The set of points on the plane With INTEGER coordinates z 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity called 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

  19. VECTORS SCALARS EXAMPLE 3

  20. VECTORS This is a commutative group SCALARS

  21. VECTORS SCALARS This is a field

  22. VECTORS For any two scalars aand b and for any vector: ( a b ) = a ( b ) For any scalar s and any vector: sV For any scalar sand for any two vectors: s ( ) = s s For any vector : 0 = ( ) ( ) v w v w z z v For any two scalars aand b and for any vector: ( a +b ) = a b For any vector : 1 = v v ( ) ( ) 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.

  23. VECTORS For any scalar s and any vector: sV For any vector : 0 = z z v For any vector : 1 = v v 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.

  24. VECTORS For any two scalars aand b and for any vector: ( a b ) = a ( b ) ( 2 2 ) A = 2 ( 2 A) ( 1 ) A = 2 ( B) 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. A A

  25. VECTORS For any two scalars aand b and for any vector: ( a b ) = a ( b ) For any scalar s and any vector: sV For any scalar sand for any two vectors: s ( ) = s s For any vector : 0 = ( ) ( ) v w v w z z v For any two scalars aand b and for any vector: ( a +b ) = a b For any vector : 1 = v v ( ) ( ) 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.

  26. VECTORS For any scalar sand for any two vectors: s ( ) = s s ( ) ( ) v w v w 2 ( A B ) = 2 A 2 B ( ) ( ) 2 ( C ) = 2 A 2 B B A = C ( ) ) 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.

  27. VECTORS ( 1 +2 ) = 1 2 ( ) ( ) B B B ( 0 ) = ( ) ( ) A B B C C For any two scalars aand b and for any vector: ( a +b ) = a b ( ) ( ) 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.

  28. VECTORS For any two scalars aand b and for any vector: ( a b ) = a ( b ) For any scalar s and any vector: sV For any scalar sand for any two vectors: s ( ) = s s For any vector : 0 = ( ) ( ) v w v w z z v For any two scalars aand b and for any vector: ( a +b ) = a b For any vector : 1 = v v ( ) ( ) 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.

  29. For any two scalars aand b and for any vector: ( a b ) = a ( b ) For any scalar s and any vector: sV For any scalar sand for any two vectors: s ( ) = s s For any vector : 0 = ( ) ( ) v w v w z z v For any two scalars aand b and for any vector: ( a +b ) = a b For any vector : 1 = v v ( ) ( ) EXAMPLE 4 V= the set of all functions that are continuous on the interval [ 0 , 1 ] symbolized C [ 0 , 1 ] real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identityf(x) = 0 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.

  30. 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 identityf(x) = 0

  31. For any two scalars aand b and for any vector: ( a b ) = a ( b ) For any scalar s and any vector: sV For any scalar sand for any two vectors: s ( ) = s s For any vector : 0 = ( ) ( ) v w v w z z v For any two scalars aand b and for any vector: ( a +b ) = a b For any vector : 1 = v v ( ) ( ) EXAMPLE 5 V= the set of all points on any line through the origin. In particular, the points on y = 2x real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity 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.

  32. The inverseof is . V= the set of all points on any line through the origin. In particular, the points on y = 2x real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity 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. closure + =

  33. V= the set of points on a line through the origin. V is a vector space. V is a SUBSET of another vector space R2. V is called a SUBSPACE of R2.

  34. 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 arith- metic A set of objects called SCALARS with operations + and × 4 7 -1 ¾ 19 -67 45 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. NO closure + =

  35. EXAMPLE 6 V= the set of all points on any plane through the origin. In particular, the points on 2x + 3y - z = 0 real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity 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. closure: and are both points on the given plane. The sum is also a point on the plane.

  36. V= the set of all points on any plane through the origin. real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity 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. closure: More generally, the points on any plane through the origin must satisfy an equation of the form: ax + by + cz = 0 if then

  37. EXAMPLE 7 V= the set of all solutions to a HOMOGENIOUS system of linear equations real numbers ordinary arith- metic A set of objects called SCALARS with operations + and × 4 7 -1 ¾ 19 -67 45 ½ The vectors form a COMMUTATIVE GROUP with an identity 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. .

More Related