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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 ×.

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Two sets:VECTORS and SCALARS four operations:

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  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. .

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