## KonradZuse Group Title Anyone good with inner products and their axioms? one year ago one year ago

1. tyteen4a03 Group Title

2. KonradZuse Group Title

5. Please refer to Denition 1 on page 335 of the textbook. Given u = (u1; u2; u3) and v = (v1; v2v3). Determine whether or not the following are inner products on R3. For those that are not, list the axioms that do not hold.

3. KonradZuse Group Title

(a) (u,v) = u[1] v[1] + u[3] v[3] (b) (u,v) = 2 2 2 2 2 2 u[1] v[1] + u[2] v[2] + u[3] v[3] (c) (u,v) = 2 u[1] v[1] + u[2] v[2] + 4 u[3] v[3] (d) (u,v) = u[1] v[1] - u[2] v[2] + u[3] v[3]

4. KonradZuse Group Title

I'm not 100% sure about Axiom 4, but it seems like so far 1 and 2 satify them all... thoughts? @tyteen4a03

5. KonradZuse Group Title

@Hero save me.

6. Hero Group Title

Bro, I took linear algebra already, but I'd have to re-familiarize myself with this.

7. Hero Group Title

Hang on for a minute

8. Hero Group Title

@KonradZuse, where in the book is that question?

9. KonradZuse Group Title

not in the book.

10. KonradZuse Group Title

sorry it's kind of screwed up in formatting.

11. KonradZuse Group Title

1 second.

12. KonradZuse Group Title

13. KonradZuse Group Title

I posted part of my answer of part b, as well as part c and d. I can post everything if oyu wihs.

14. KonradZuse Group Title

okay hold on.

15. KonradZuse Group Title

here happy?

16. KonradZuse Group Title

I just want to know if I'm doing this correctly/ what exactly is axiom 4 asking?

17. KonradZuse Group Title

saying*

18. KonradZuse Group Title

@UnkleRhaukus

19. Zarkon Group Title

I guess I'll reply...a lot of your work is not correct..I'll let you think about that for a while

20. UnkleRhaukus Group Title

where are the axioms you are referencing ?

21. KonradZuse Group Title

one second.

22. KonradZuse Group Title

23. UnkleRhaukus Group Title

(a) $(\textbf u,\textbf v)=u_1 v_1+u_3v_3$

24. KonradZuse Group Title

I was thinking that might fail because of the new u2v2, but not too sure?

25. UnkleRhaukus Group Title

$(\textbf u,\textbf v)=u_1 v_1+u_3v_3=u_3v_3+u_1 v_1=(\textbf v,\textbf u)$ the symmetry axiom holds

26. KonradZuse Group Title

ok.

27. KonradZuse Group Title

I actually posted my work up a bit. Should I repost it?

28. KonradZuse Group Title

I did a and B I just was confused about axiom 4, and so far it doesn't seem like anything is going to fail. Thoughts?

29. UnkleRhaukus Group Title

$(\textbf u+\textbf v,\textbf w)=(\textbf u+\textbf v)_1\textbf w_1+(\textbf u+\textbf v)_3\textbf w_3$$\qquad\qquad\quad=u_1w_1+v_1w_1+u_3w_3+v_3w_3$$\qquad\qquad\quad=u_1w_1+u_3w_3+v_1w_1+v_3w_3$$\qquad\qquad\quad=(\textbf u,\textbf w)+(\textbf v,\text w)$ the additivity axion holds

30. UnkleRhaukus Group Title

$(k\textbf u,\textbf v)=ku_1 v_1+ku_3v_3$$\qquad\quad=k(u_1 v_1+u_3v_3)$$\qquad\quad=k(\textbf u,\textbf v)$ the homogeneity axiom holds

31. KonradZuse Group Title

32. KonradZuse Group Title

not sure what was wrong with my work. I did exactly what the book/my other homework did for it.

33. UnkleRhaukus Group Title

$(\textbf v,\textbf v)=v_1v_1+v_3v_3$$\qquad\quad=v_1^2+v_3^2≥0$ $(\textbf v,\textbf v)=0\iff \textbf v=\textbf 0$ the positivity axiom holds

34. Zarkon Group Title

that is the problem..axiom 4

35. Zarkon Group Title

your work done on axiom 4 is not correct

36. Zarkon Group Title

remember $$u$$ and $$v$$ are 3-dimensional vectors

37. Zarkon Group Title

let $$u=v=<0,1,0>$$ what is $$(u,v)$$

38. Zarkon Group Title

or same thing ...what is $$(v,v)$$

39. KonradZuse Group Title

I will check it out.

40. KonradZuse Group Title

I guess I'm still confused how to prove axiom 4....

41. Zarkon Group Title

you don't prove axiom 4...it is an axiom

42. KonradZuse Group Title

errr "satisfy axiom 4"

43. Zarkon Group Title

(a) is not an inner product...I gave you a counter example

44. Zarkon Group Title

((0,1,0),(0,1,0))=0*0+0*0=0 but (0,1,0) is not the zero vector

45. KonradZuse Group Title

ic so axiom 4 isn't = 0 thus fails?

46. Zarkon Group Title

it fails... $$(v,v)=0\Leftrightarrow v=\vec{0}$$

47. UnkleRhaukus Group Title

i see what you are saying now @Zarkon $(\textbf v,\textbf v)=v_1v_1+v_3v_3$$\qquad\quad=v_1^2+v_3^2≥0$$(\textbf v,\textbf v)=0\iff \textbf v=(0,v_2,0)\not\equiv\textbf 0$the positivity axiom does not hold in the general case

48. Zarkon Group Title

correct...but one only needs so show a single counter example to show that it is not an inner product (i'm lazy that way :) )

49. KonradZuse Group Title

I'm so confused though :)

50. KonradZuse Group Title

I guess I can see how (0,1,0)(0,1,0) = 1 not 0. So all we have to do is multiply and see if = 0?

51. UnkleRhaukus Group Title

how did you get = 1 ?

52. KonradZuse Group Title

I thought that's what he said above, I guess I'm just too konfused hahaha :(

53. UnkleRhaukus Group Title

the rule was; to add the product of the first components to the product of the third components , the rule satisfied the 4th axiom iff the rule only generates zero when the vector is the zero vector. Zarkon has shown that that the second component of the vector could be anything , therefore there are many vectors that generate zero , not just the zero vector)

54. KonradZuse Group Title

hmm ic.

55. KonradZuse Group Title

So does Axiom 4 only fail for the first one?

56. Zarkon Group Title

another one of your problems fails axiom 4

57. KonradZuse Group Title

hmm time to find it, and see if any other's fail anything....

58. KonradZuse Group Title

it seems like the others pass the other axioms, now to try and figure out which one fails #4 :P

59. KonradZuse Group Title

Does D by any chance fail Axiom 2?

60. Zarkon Group Title

no...but it fails another axiom

61. Zarkon Group Title

I'll tell you that only one of the four is an inner product

62. KonradZuse Group Title

geh..

63. KonradZuse Group Title

C also seems to fail one.

64. KonradZuse Group Title

my program just crashed so h/o.

65. KonradZuse Group Title

all of my work is gone, I'm flipping out, hold on please.

66. KonradZuse Group Title

SO d fails some axiom, but what about C @Zarkon ?

67. Zarkon Group Title

(c) is an inner product...the only one

68. KonradZuse Group Title

Damn now I got to look into b... Maybe that also fails axiom 4. D must fail axiom 3, maybe 4, IDk. Need to rework it, but flipping out trying to finish these other problems. Thanks for all the help I really appreciate it!

69. Zarkon Group Title

$(u,v)=u_1^2v_1^2+u_2^2v_2^2+u_3^2v_3^2$ $(ku,v)=(ku_1)^2v_1^2+(ku_2)^2v_2^2+(ku_3^2)v_3^2$ $=k^2(u_1^2v_1^2+u_2^2v_2^2+u_3^2v_3^2)=k^2(u,v)$

70. KonradZuse Group Title

axiom 3

71. KonradZuse Group Title

do you think you could help me with my other problems? I would greatly appreciate it.