Given 3 simultaneous equations with 3 unknowns each, how do i determine if they have 0, 1, or infinite sets of solutions? I know how to convert the equations into reduced echelon form, if it helps. Thanks.
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Anonymous2008-09-24 7:32
Rewrite one equation in terms of one of the unknowns. Substitute that into the second equation, and rewrite it in terms of another unknown. Substitute into the third and solve.
If it leads to a contradiction they have no solutions. If it doesn't work they have an infinite number of solutions. Otherwise they have one unique solution.
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Anonymous2008-09-24 9:11
Write as a matrix.
If it has non-zero determinant, it has an inverse, and then you have your solutions.
Fucked if I can be arsed to remember the difference between how many solution it has though.
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Anonymous2008-09-25 12:42
If any equation is a multiple of another, there are infinite solutions as they are two identical planes
If any of the equations is multiple of another's "ax+by+cz" part, but not the constant part, then there are 0 solutions as they are non intersecting planes.
Otherwise, 1 solution.
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Anonymous2008-09-27 6:29
Think about it GEOMETRICALLY ASSHAT
You have 3 planes.
If the three normals are all unique then you have one solution
If there are two or more normals that are the same you have infinite or zero solutions.
To see if there are infinite solutions check if the planes are identical.
If they are not identical than there are no solutions.
>>6
"Using legs to walk means you don't have to think about the geodesic course in which you will decide to follow today. Where's your path of least action when gravity's holding you firmly to the ground? That's right, fucking with your legs."
>>10
That "then let N go to 9" joke isn't entirely unfounded in reality, you know. People with a Ph.D in mathematics often have had their head so well and truly fucked that they won't even feel N-dimensional problems and their limp little dicks.
Anyway, who knows whether they're thinking of linear algebra geometrically or just reducing geometrics to linear algebra. I'm not a mathematician myself, I just know enough of them to appreciate the mathematician jokes.
Doing my third year of a maths degree, and I'm not thinking of linear algebra geometrically.
I do still think of open sets as little balls though, which is a terrible habit when the space isn't metrizable :(
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Anonymous2008-09-29 20:23
Okay I am more into game programming so thinking of everything geometrically makes more sense to me.
Even in N dimensions it is still intersections/lack of intersections.
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Anonymous2008-10-24 5:33
>>1
Set up matrix and row reduce. If you can get row reduced echelon form, you have one unique solution (actually, it's referred to as a one-to-one linear transformation). If you get a free variable, you have infinite solutions. Not sure how to prove an inconsistent system without given solutions (b-vector for the Ax = b system).
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Anonymous2008-10-30 16:57
>>6
How is that so complicated? The three axis changing over another axis, and the change in that over another axis. I had to construct 5 dimensional cross sections for my introductory physics class.
Yeah, I can talk about it too, visualising it's not easy.
Also, fine, n dimensions, whatever. Pick the smallest n you can't deal with, now you may as well use linear algebra, so you may as well use it for the rest.
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Anonymous2008-11-01 6:51
[ \left[
\begin{array}{ c c }
1 & 0 \\
0 & 1
\end{array} \right]
\]
has 1 unique solution. [ \left[
\begin{array}{ c c }
1 & 0 \\
1 & 0
\end{array} \right]
\]
has infinitely many solutions.
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Anonymous2008-11-01 6:54
\left[
\begin{array}{ c c }
1 & 0 \\
0 & 1
\end{array} \right]
has 1 unique solution. \left[
\begin{array}{ c c }
1 & 0 \\
1 & 0
\end{array} \right]