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