anonymous
  • anonymous
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.
Mathematics
  • Stacey Warren - Expert brainly.com
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chestercat
  • chestercat
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rvc
  • rvc
do you know matrices?
tkhunny
  • tkhunny
Have you considered a determinant?
anonymous
  • 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.

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rvc
  • rvc
equations are consistent if the determinant equals to zero
rvc
  • rvc
hello :)
anonymous
  • 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. -_-;
rvc
  • rvc
do you remember how to find determinant?
anonymous
  • anonymous
No, not really. I'm sorry.
anonymous
  • anonymous
I missed it. How do I find the determinant?
rvc
  • rvc
|dw:1441072730208:dw|
anonymous
  • anonymous
Hm... So how does that translate into my word problem?
anonymous
  • anonymous
if we break it down into a matrices 2 4 4 4 0 1 2 -2 2 3 0 2
anonymous
  • 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
rvc
  • 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. :)
anonymous
  • anonymous
@tkhunny
dan815
  • dan815
hey have you figured out how to take a determinant yet
anonymous
  • anonymous
Nope, I have not. Do you have any time to help me figure out how to do it?
dan815
  • dan815
yeah
anonymous
  • 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.
dan815
  • dan815
okay basically to start off, you have an equation which is a linear function of 3 variables
dan815
  • dan815
for example 1x+2y+3z=5 lets say this equation instead of x1,x2,x3
anonymous
  • anonymous
Alright
dan815
  • dan815
have you seen equations like this before its the qeuation of a plane in 3d space
anonymous
  • anonymous
Yes, a long time ago. It's been a while since I've had to work through anything like this though.
dan815
  • dan815
okay i see well dont worry
dan815
  • dan815
now how about a gradient have you heard about gradients or normal vectors to planes
anonymous
  • anonymous
Nope. I have not dealt with that. We're just starting out the school year with matrices.
dan815
  • 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
dan815
  • 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
anonymous
  • anonymous
That's okay! I appreciate your willingness to explain!
anonymous
  • anonymous
That makes sense.
dan815
  • 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
dan815
  • dan815
so your matrix is the constants of all the coefficceints in order
anonymous
  • anonymous
Alright. So we break it up into a matrix... 2 4 4 4 0 1 2 -2 2 3 0 2
dan815
  • dan815
|dw:1441074412998:dw|
dan815
  • dan815
we dont care about the constants in this case
anonymous
  • anonymous
Alright, so just the first three columns then.
dan815
  • dan815
yeah the numbers infront of variables only
anonymous
  • anonymous
2 4 4 0 1 2 2 3 0
dan815
  • dan815
now to take a determinant of a matrix
dan815
  • dan815
first u learn to take the determinant of 2 by 2 matrix
anonymous
  • anonymous
How do you take the determinant.
dan815
  • dan815
|dw:1441074586338:dw|
dan815
  • dan815
this is the determinnat of 2 by 2 matrix
dan815
  • dan815
difference* of the product of the diagonals in that order
dan815
  • dan815
we will use this to take the determinant of 3by3 matrix
anonymous
  • 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?
dan815
  • dan815
|dw:1441074679006:dw|
dan815
  • dan815
do get the mini determinants from the big determinant what you do is cover this way
dan815
  • dan815
|dw:1441074913573:dw|
dan815
  • dan815
|dw:1441074951428:dw|
dan815
  • dan815
https://www.youtube.com/watch?v=ROFcVgehEYA
dan815
  • dan815
just watch that its a lot better
dan815
  • dan815
since the determinant is non zero you can conclude there will be only 1 intersection as they are all independant
dan815
  • dan815
because if u look at 3 planes
anonymous
  • 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!
dan815
  • dan815
|dw:1441075204298:dw|
dan815
  • dan815
now if u throw in another plane, that means u are really checking the intersection of a line and a plane
dan815
  • dan815
so if that plane is independant that means it will be a line passing through a plane so just a point
dan815
  • 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
anonymous
  • anonymous
Okay!
dan815
  • dan815
just know the formal definition of independance
dan815
  • dan815
wait
dan815
  • dan815
http://prntscr.com/8b85p2
anonymous
  • 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.
dan815
  • 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
anonymous
  • 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.
dan815
  • 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
anonymous
  • anonymous
That makes sense.
dan815
  • dan815
yep basically the definition in there is the same thing not stated for all sizes of vectors
dan815
  • 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
dan815
  • dan815
if that is the only solution, meaning taking nothing to get nothing, everything else doesnt product 0 that means they are lienarly independant
dan815
  • 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
dan815
  • 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,
dan815
  • 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
anonymous
  • anonymous
Neat! Thank you!
dan815
  • dan815
you're welcome
dan815
  • dan815
good luck with this stuff
anonymous
  • 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.

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