## anonymous one year ago Do the three lines 2x1 + 4x2 + 4x3 = 4, x2 - 2x3 = -2, and 2x1 + 3x2 = 0 have at least one common point of intersection? It's linear algebra, and has to do with determining the consistency of systems. Like a matrices problem, the 1 2 and 3 represent rows, not exponents.

1. rvc

do you know matrices?

2. tkhunny

Have you considered a determinant?

3. anonymous

Hi rvc, I know a little about matrices since we just started going over them, but this one has thrown me for a loop.

4. rvc

equations are consistent if the determinant equals to zero

5. rvc

hello :)

6. anonymous

Okay, so how do I make that happen? I was sick and missed the lecture, so I'm a bit lost. I sorry for bothering you. -_-;

7. rvc

do you remember how to find determinant?

8. anonymous

No, not really. I'm sorry.

9. anonymous

I missed it. How do I find the determinant?

10. rvc

|dw:1441072730208:dw|

11. anonymous

Hm... So how does that translate into my word problem?

12. anonymous

if we break it down into a matrices 2 4 4 4 0 1 2 -2 2 3 0 2

13. anonymous

But I'm not quite sure I understand how the other numbers within the matrices are broken up to be multiplied(?) by either a11 a12 or a13

14. rvc

atm i need to go. i have my exam in the next hour. all the best and sorry to leave you in between this problem. i m sure @mathmate will help you. :)

15. anonymous

@tkhunny

16. dan815

hey have you figured out how to take a determinant yet

17. anonymous

Nope, I have not. Do you have any time to help me figure out how to do it?

18. dan815

yeah

19. anonymous

Wow! That'd be great. I'm trying to work through the problems in the book for the lecture I missed but I'm kind of lost.

20. dan815

okay basically to start off, you have an equation which is a linear function of 3 variables

21. dan815

for example 1x+2y+3z=5 lets say this equation instead of x1,x2,x3

22. anonymous

Alright

23. dan815

have you seen equations like this before its the qeuation of a plane in 3d space

24. anonymous

Yes, a long time ago. It's been a while since I've had to work through anything like this though.

25. dan815

okay i see well dont worry

26. dan815

27. anonymous

Nope. I have not dealt with that. We're just starting out the school year with matrices.

28. dan815

okay well then for now i cant fully explain to you why this matrix method is working, but lets just go through the determinant

29. dan815

a non zero determinant means that all 3 of your systems are independant of each other meaning they are not parallel or one of the system is not a combination of the 2 remaining systems

30. anonymous

That's okay! I appreciate your willingness to explain!

31. anonymous

That makes sense.

32. dan815

okay we dont really care about the constant term for now, again this has something do with the normal vectors how they wont depend on the constant

33. dan815

so your matrix is the constants of all the coefficceints in order

34. anonymous

Alright. So we break it up into a matrix... 2 4 4 4 0 1 2 -2 2 3 0 2

35. dan815

|dw:1441074412998:dw|

36. dan815

we dont care about the constants in this case

37. anonymous

Alright, so just the first three columns then.

38. dan815

yeah the numbers infront of variables only

39. anonymous

2 4 4 0 1 2 2 3 0

40. dan815

now to take a determinant of a matrix

41. dan815

first u learn to take the determinant of 2 by 2 matrix

42. anonymous

How do you take the determinant.

43. dan815

|dw:1441074586338:dw|

44. dan815

this is the determinnat of 2 by 2 matrix

45. dan815

difference* of the product of the diagonals in that order

46. dan815

we will use this to take the determinant of 3by3 matrix

47. anonymous

Okay. How would you combine a 3 by 3 though? A B C D E F G H I... _> AE-BD, BF-CE, DH-EG, EI-FH?

48. dan815

|dw:1441074679006:dw|

49. dan815

do get the mini determinants from the big determinant what you do is cover this way

50. dan815

|dw:1441074913573:dw|

51. dan815

|dw:1441074951428:dw|

52. dan815
53. dan815

just watch that its a lot better

54. dan815

since the determinant is non zero you can conclude there will be only 1 intersection as they are all independant

55. dan815

because if u look at 3 planes

56. anonymous

Wow... Thank you so much! That's amazing! It makes a lot of sense, way more than I thought it would. This is a great addition to my notes!

57. dan815

|dw:1441075204298:dw|

58. dan815

now if u throw in another plane, that means u are really checking the intersection of a line and a plane

59. dan815

so if that plane is independant that means it will be a line passing through a plane so just a point

60. dan815

if the plane was not independant that means the line also belongs in that plane, which means the plane stated is actually redundant which is why ud get a 0 determinant, that wont really make sense yet but ya something to keep in mind for later

61. anonymous

Okay!

62. dan815

just know the formal definition of independance

63. dan815

wait

64. dan815
65. anonymous

Okay, what else would you like to explain. I'm not leaving until you're finished ^_^. I just "fanned" you and wrote what I hope will be a glowing testimonial of your efforts.

66. dan815

basically this a bunch of vectors are said to be linearly independant, if there is no way to do add them or some factor of them up together to end up with 0

67. anonymous

I will make sure to add the formal definition of independence in my notes as well, so that I remember it for the next lecture.

68. dan815

for example v1,v2 linearly independant if av1+bv2=0 only happens when both a and b are 0 which is the trivial case, or basically taking nothing to get nothing

69. anonymous

That makes sense.

70. dan815

yep basically the definition in there is the same thing not stated for all sizes of vectors

71. dan815

you can have v1,v2,v3...vn and they all independant if a1*v1+a2*v2+..._an*vn=0 the only solution is that a1,a2,a3...an are all 0

72. dan815

if that is the only solution, meaning taking nothing to get nothing, everything else doesnt product 0 that means they are lienarly independant

73. dan815

it is is easy to see somthing like this when u talk about orthogonal vectors which are vectors than are as indepednant as possible of each other

74. dan815

for example if u take a vector in the x direction and another vector in the y direction, theres no way to add them to get 0,

75. dan815

once u have the existance of a vector in x, how can you even add some vector in the y direction to make it go back to 0

76. anonymous

Neat! Thank you!

77. dan815

you're welcome

78. dan815

good luck with this stuff

79. anonymous

Thank you! If nothing else, this experience has taught me to never get sick ever again. That will be the day lecture will be over something I don't understand.