2. Two sets:VECTORS and SCALARS four operations:

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 z 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. 2 =

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 z 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. 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

5. 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 × 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. 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 arith- metic

6. 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 × 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. For any scalar s and any vector: s  V V= The set of points on the plane: real numbers ordinary arith- metic 2 s s s

7. 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 × 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. 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 arith- metic

8. 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 × 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. V= For any vector : 0 = z v z The set of points on the plane: real numbers ordinary arith- metic 0

9. 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 × 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. 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 arith- metic

10. 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 × 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. For any vector : 1 = v v V= The set of points on the plane: real numbers ordinary arith- metic 1

11. 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 × 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. 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 arith- metic

12. 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 × 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. 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 arith- metic ( 4 × 3 )= 43 ( ) ( 12 )= 4 ( )

13. 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 × 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. 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 arith- metic

14. 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 × 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. 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 arith- metic 3 ( ) = 3 3 ( ())

15. 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 × 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. 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 arith- metic

16. 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 × 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. 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 arith- metic ( 2 + 3 ) = 2 3 (())

17. 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 × 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. 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 arith- metic

18. 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 × 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. 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 arith- metic With INTEGER coordinates

19. 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 × 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. For any scalar s and any vector: s  V V= The set of points on the plane real numbers ordinary arith- metic With INTEGER coordinates ½ Could you use integers rather than real numbers as your set of scalars?

20. VECTORS SCALARS

21. VECTORS SCALARS This is a commutative group

22. VECTORS SCALARS This is a field

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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 ) = 1 2 (()) BBB ( 0 ) = (()) BB A C C

29. 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

30. The vectors form a COMMUTATIVE GROUP with an identity f(x) = 0 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. 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 arith- metic

31. 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 ]

32. The vectors form a COMMUTATIVE GROUP with an identity 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. 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 arith- metic

33. The vectors form a COMMUTATIVE GROUP with an identity 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. V= the set of all points on any line through the origin. In particular, the points on y = 2x real numbers ordinary arith- metic += closure The inverse of is.

34. 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.

35. 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. 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 += NO closure

36. The vectors form a COMMUTATIVE GROUP with an identity 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. 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 closure: and are both points on the given plane. The sum is also a point on the plane.

37. The vectors form a COMMUTATIVE GROUP with an identity 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. V= the set of all points on any plane through the origin. real numbers ordinary arith- metic closure: ax + by + cz = 0 More generally, the points on any plane through the origin must satisfy an equation of the form: if then

38. The vectors form a COMMUTATIVE GROUP with an identity 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. V= the set of all solutions to a HOMOGENIOUS system of linear equations real numbers ordinary arith- metic.