Three matchbox cars are traveling at different speeds and directions: 10 m/s northeast, 10 m/s southeast, and 15 m/s northeast. The document discusses using matchbox cars and their speeds and directions as an example to introduce concepts related to vectors, vector spaces, and tensors. It describes how the speeds of the cars can be added and multiplied together, forming a basis set and elements within a vector space to represent and operate on the cars and their motion.

2. • A MATCHBOX CAR GOIMG 10 M/S AT NORTH EAST
• A MATCHBOX CAR GOIMG 10 M/S AT SOUTH EAST
• A MATCHBOX CAR GOIMG 15 M/S AT NORTH EAST

5. Color box
The color can be inferred from the position

7. Similarily – a Hardness box

8. • Which again-has 3 apperatures-an input
• And hard electrons come out this port and soft elecrtron come out
this port

9. And the workings inside the box do not matter

10. You can imagine-so little worker inside the
box deciding what is extracted

11. The box can be built with dwarfs, elves, silver
atoms, atoms with a lot of neutrons, etc

12. But what ever they are built with-they give the
same results the professor is gonna describe

13. So-a chief property of the hardness and color
boxes is the ‘results are repeatable”

14. If I send in an electron and it comes out a
color box -black

15. Then, I send it in another color box-and it
comes out black again

16. So-in a diagram-if I say an “IN”

17. And –I discover that it comes out the white
apparature

18. And –I take the ones that come out the white
apperature –and I send them into a color box

19. With 100% confidence-meaning 100% of the
time

20. The electron coming out the white port and
into the color box

21. Will come out the white apperature again

22. And zero percent of the time

23. Where to come out the black apperature

24. So –this is a persistent property

25. You notice that it is white-and it is still white

27. Dealing with hardness-if I send a bunch of
electrons in a hardness box

28. Here is the important thing

29. And I am going to isolate the ones that come
out soft

43. You have these images in google

44. What Is A Tensor Lesson #1: Elementary vector
spaces

45. I used to have a car set . .

46. WOULD BRAGG –COME OVER TO RACE
OTHER KIDS

47. And –some were faster than others
• A MATCHBOX CAR GOIMG 10 M/S AT NORTH EAST
• A MATCHBOX CAR GOIMG 10 M/S AT SOUTH EAST
• A MATCHBOX CAR GOIMG 15 M/S AT NORTH EAST

48. LETS CALL THEM –THE VECTORS

49. SOME SPACE—SPEDS- A COLLECT OF SPEEDS-
A TENSOR

50. A VECTOR SPACE-THE BOX OF CARS

51. WE CAN ADD BVELOCITIES OF THE CARS

52. WE CAN MULTIPLY THE- AND AND THE
SPEEDS-THEY HAVE THE SAME UNITS

53. THEY FORM A BASIS SET

54. NOW-LET’S SAY ROB –PETER AND ERIC ARE
GONNA RACE THEIR CARS-”TRUE STORY”

55. 2 VECTORS CAN ADD

56. AND I SAID WE CAN ADD 10 M/S TO 20 M/S
TO MAKE 30 M/S GIONG NORTH

57. YOU CAN MULTIPLY THEM-THIS IS CALLED
SCALAR MULTIPLICATIO

58. WHEN YOU DO THIS-IT BECOMES AN
ELEMENT OF THE VECTOR SPACE
• LET ME RUN THIS BY YOU

59. THIS IS A “VECTOR IN W”

60. THE PROCESS IS CALLED SCALAR
MULTIPLICATION

61. THE OBJECTS THAT COME OUT OF THE REAL
NUMBERS ARE CALLED SCALARS

62. THE TENSOR-COULD BE A BOSX SHAPE-BUT
ALSO COULD BE SHAPED LIKE ABALL

66. • Linear Algebra Example Problems - Vector Space Basis Example #2
• https://www.youtube.com/watch?v=vejkdUigyIg

67. • How to Prove a Set is a Subspace of a Vector Space
• https://www.youtube.com/watch?v=Eawc_ZuQI_8