3. 60 10 20 30 40 50 A man drops a ball from the top of a building.

4. 60 10 20 30 40 50 After ½ second, the ball has fallen 4 feet. 1/2 4

5. 60 10 20 30 40 50 60 10 20 30 40 50 After 1 second, the ball has fallen 16 feet. 116 1/2 4

6. 60 10 20 30 40 50 60 10 20 30 40 50 After 3/2 second, the ball has fallen 36 feet. 116 36 1/2 4 60 10 20 30 40 50 3/2

7. 60 10 20 30 40 50 60 10 20 30 40 50 After 2 seconds, the ball has fallen 64 feet. 116 36 264 1/2 4 60 10 20 30 40 50 3/2 60 10 20 30 40 50

8. 60 10 20 30 40 50 116 36 264 1/2 4 3/2 Motion is described as a set of ordered pairs. { ( 1/2, 4 ), ( 1,16 ), ( 3/2, 36 ), ( 2, 64 ) }

9. 60 10 20 30 40 50 116 36 264 1/2 4 3/2 Motion is described as a set of ordered pairs. { ( 1/2, 4 ), ( 1,16 ), ( 3/2, 36 ), ( 2, 64 ) } Sometimes there is a pattern, and we can write an equation: d = 16 t 2 t is time in seconds d is distance in feet

10. 60 10 20 30 40 50 116 36 264 1/2 4 3/2 More generally, a function is defined as a set of ordered pairs. { ( 1/2, 4 ), ( 1,16 ), ( 3/2, 36 ), ( 2, 64 ) } When we write an equation for a function, the solutions (ordered pairs) define the function. d = 16 t 2 t is time in seconds d is distance in feet

11. 60 10 20 30 40 50 116 36 264 1/2 4 3/2 A function is defined as a set of ordered pairs. { ( 1/2, 4 ), ( 1,16 ), ( 3/2, 36 ), ( 2, 64 ) } The DOMAIN of the function = { 1/2, 1, 3/2, 2 } The RANGE of the function = { 4,16, 36, 64 }

12. 60 10 20 30 40 50 116 36 264 1/2 4 3/2 The DOMAIN of the function = 1/2 1 3/2 2 The RANGE of the function = 4 16 36 64 The function is a mapping that relates every Domain element t to a unique corresponding Range element, denoted f(t) and called the image of t

13. 60 10 20 30 40 50 116 36 264 1/2 4 3/2 The DOMAIN of the function = 1/2 1 3/2 2 The RANGE of the function = 4 16 36 64 The function is a mapping that relates every Domain element t to a unique corresponding Range element, denoted f(t) and called the image of t 4 is the image of ½ 16 is the image of 1 36 is the image of 3/2 64 is the image of 2 4 = f ( ½ ) 16 = f ( 1 ) 36 = f ( 3/2 ) 64 = f ( 2 )

14. 60 10 20 30 40 50 116 36 264 1/2 4 3/2 The DOMAIN of the function = 1/2 1 3/2 2 The RANGE of the function = 4 16 36 64 The function (of high school algebra fame) relates a set of real numbers to another set of real numbers. Next we will examine a mapping that links a set of vectors to another set of vectors. In doing so, we use much of the same terminology that we used in the study of functions. A function is a type of mapping.

16. A farmer plans to purchase a herd of cows. He considers 2 breeds: Purple cows and Brown cows

17. Each day a purple cow will eat 1 bale of hay and will produce 2 bottles of milk Each day a brown cow will eat 2 bales of hay and will produce 3 bottles of milk

18. purple brown A herd comprised of 50 purple and 70 brown cows will consume 190 bales of hay and produce 310 bottles of milk.

19. purple brown A herd comprised of 100 purple and 30 brown cows will consume 160 bales of hay and produce 290 bottles of milk.

20. purple brown A herd comprised of 80 purple and 150 brown cows will consume 380 bales of hay and produce 610 bottles of milk.

21. purple brown The DOMAIN of the mapping: These vectors describe the composition of the herd, and this determines

22. purple brown The DOMAIN of the mapping: These vectors describe the composition of the herd, and this determines The RANGE of the mapping: These vectors describe the daily food intake and milk yield.

23. purple brown Just as it is sometimes possible to find an equation to define a function, it is sometimes possible to produce a matrix to define a vector space mapping.

24. purple brown Just as it is sometimes possible to find an equation to define a function, it is sometimes possible to produce a matrix to define a vector space mapping. # bales of hay = 1 (# purple cows) + 2 (# brown cows)

25. purple brown Just as it is sometimes possible to find an equation to define a function, it is sometimes possible to produce a matrix to define a vector space mapping. # bales of hay = 1 (# purple cows) + 2 (# brown cows) # bottles of milk = 2 (# purple cows) + 3 (# brown cows)

26. purple brown Just as it is sometimes possible to find an equation to define a function, it is sometimes possible to produce a matrix to define a vector space mapping. # bales of hay = 1 (# purple cows) + 2 (# brown cows) # bottles of milk = 2 (# purple cows) + 3 (# brown cows)

27. purple brown eg:

28. purple brown eg:

29. purple brown For every domain element ( a vector in R 2 whose entries are the numbers of each breed of cow) there is a unique corresponding range element ( a vector in R 2 whose entries are the numbers of bales consumed and bottles produced.) A

30. purple brown A eg: