Google says that we need to find the pivots of the system matrix, and the variables represented by those pivots are the basic variables. In contrast, my textbook says that we can choose any variables to be the basic ones as long as, when we put all of them on the left side and everything else on the right side, the resulting matrix of coefficients has a determinant different from 0 (or a more useful way to think about it is, its rank is the same as the number of variables excluding the ones we put on the right side).
So because I don't know which is correct, I decided to try an example problem. The textbook provides an example solution to this problem, where it decides that x1, x2 and x4 are the basic variables. The final result is two tuples, (3, -2, 1, 0, 0) and (5, -3, 0, 0, 1). Supposedly any linear combination of these two tuples will give a tuple which represents a solution to the system I linked to.
I decided to try the same problem, but chose x3, x4 and x5 as the basic variables, because this still obeys the rule I mentioned in the first paragraph. Going through the same process, I ended up with two tuples which are different: (1, 0, -3, 0, 2) and (0, 1, -5, 0, 3). I don't know how to prove whether these two tuples are "equivalent" to those other two from the textbook example, but trying a few examples by hand, any tuple that I could write as a linear combination of the first two tuples, I could also write as a linear combination of the second two.
So, which is it? Is my textbook correct?
-----
Edit: After thinking about it some more, I think one explanation might be that this only worked because in the first example, x1 and x2 are written in terms of x3 and x5, while as in my attempt, I went vice-versa (x4 is equal to 0 so it didn't matter). Could this be the case, that you can swap the basic and free variables, but you can't "mix" them?
I am studying Linear Algebra, and the first lesson said that matrices are a shorthand notation for systems of linear equations. Then, in the third lesson, it says that matrices are a series of row/column vectors. I do not understand what matrices are.
For example,
2x = 5
x = 7
would be represented as a matrix like this:
[2]
[1]
or an augmented matrix
[2 | 5]
[1 | 7]
But then there is a column vector.
[2]
[1]
And apparently the 2 is the x-coordinate and now the 1 is the y-coordinate.
But in one of these matrices, the column represents x values whereas in the other it represents by x and y values.
According to my textbook, due to Cramer's rules for systems of linear equations, we know that x1 = Dx1/D, x2 = Dx2/D, etc. We can write this as:
x1 * D = Dx1, x2 * D = Dx2... xn * D = Dxn
This allows us to think about a few cases:
-
If D =/= 0, then all is well and the system has one possible set of solutions (x1, x2, x3...)
-
If D = 0 but, say, Dx1 is a non-zero number (let's say 5), then there doesn't exist a number x1 such that x1 * 0 = 5, therefore the system has no solutions
-
If D = 0 and all Dxn's are also 0, then the system is undertermined as any values of x1, x2, etc. could fulfill the equations.
At this point, my textbook mentions a caveat. It says that in rare cases it's possible to get case 3 from a system that actually has no solutions, so it would be wrong to say that it's underdetermined.
So my question is, what's the solution then? If I'm using the Cramer method and get case 3, is there a fast way to check whether the system is underdetermined or just has no solutions? Or should I just then use another method for solving it, which I could have done from the beginning.
Thanks in advance.
It gets really hard for me whenever there's any fractions involved, for example:
{(2x - 4)/3 - (y + 3)/9 = 0 {3x/8 + y/2 = -3/2
Just becomes too hard, i would like to know the steps that i should apply to solve these equations with fractions. Thanks!
hello I have a TI-84 plus CE and was wondering if there was a program I can download that can solve a system of linear equations, including systems with complex numbers.
I think I remember seeing a video about Systems of Linear Equations and how to solve them (I think using Gaussian Elimination or something like that) in a 3b1b style (definitely not someone writing on a blackboard like blackpenredpen e.g.). Unfortunately, I wasn’t able to find it anymore (it doesn’t seem like Grant himself made such a video, or I’m unable to find it). Does anyone have an idea which video it might be, or does anyone know similar good resources to learn about linear systems?
Thank you in advance!
So I'm in my first year of engineering, learning matrices. I have a test on solving systems of linear equations next monday and I wanna do well. But the problem is I get every single gaussian elimination problem (and cramers rule and inverse matrix method too for that matter, but gaussian elimination is my main struggle). I know how to solve systems of linear equations with the simultaneous equations method just fine and it always comes out right, but when I try solve it with gaussian elimination the answers are always wrong and end up as weird ass fractions like -(37/2). i've tried using the methods shown on my notes and on youtube videos for solving equations using gaussian elimination but it just never ends up working. Personally I find working within matrices very restricting, for example with simulatneous equations I can just perform simple operations to create new equations with different variables eliminated, as you do. But then with gaussian elimation I have to stay strict to the echelon form and staying within the augmented matrix and it just feels like every operation that I think is the correct one just ends up being completely incorrect. You can probably tell by now just from reading this that im not that good at math in general but this is just really pissing me off. Ideally I'd just solve all the questions in the test using simultaneous equations but my lecturer is probably gonna check that we were using matrix operations on our scripts. So please, if anyone has any tips or advice to help me crack solving gaussian elimination and solving systems of linear equations with matrices in general I would really appreciate it.
Suppose you are asked to solve {y=−2x+5
----------------------------------------------{6x+3y=15
and you end up with 0=0. How would you write the solution to this system?
Essentially it is wanting me to simply find y and x for these systems, but I have to have the work along with it. They are
A. 3x+y=9 and 7x+1y=32
B. 5x-8y=25 and -x+4y=-8
C. 2/3 x +1/4 y=18 and 1/6 x-3/8 y=-6
D. 5x4y=-14 and 3x+6y=6
Please, a ELI5 version for idiots.
I know that I have a trivial solution where x = y = z = 0. But how do I know by looking at a homogeneous system that I'll either have a trivial solution or infinite many of them?
The professor said that a homogeneous system will always have a solution. Either a trivial solution or infinite many of them.
I have a linear system of 21 equations and variables in MATLAB but I don't want to write a coefficient matrix to solve it since it's very large. Is it possible to solve the system with just the equations as input ?
https://preview.redd.it/0kehime4bgt71.jpg?width=328&format=pjpg&auto=webp&s=671f2de230c3f0e047359d9d6e2a332728fad5fe
Script:
clear
close all
clc
syms E A L l m P1 P2
GenK = [l^2 l*m -l^2 -l*m;...
l*m m^2 -l*m -m^2;...
-l^2 -l*m l^2 l*m;...
-l*m -m^2 l*m m^2];
E = 210e9;
P = 1000;
L = 1;
A = 6e-4;
%% Bar 1
theta1 = 90;
L1 = cosd(theta1);
M1 = sind(theta1);
k1 = subs(GenK,[l,m],[L1,M1])*E*A/L
k1(5,5) = 0;
k1(6,6) = 0;
k1
%% Bar 2
theta2 = 0;
L2 = cosd(theta2);
M2 = sind(theta2);
k2t = subs(GenK,[l,m],[L2,M2])
k2 = zeros(6,6);
k2(3:end,3:end) = k2t*E*A/L;
k2
%% Bar 3
theta3 = 45;
L3 = cosd(theta3);
M3 = sind(theta3);
k3t = subs(GenK,[l,m],[L3,M3])
k3=zeros(6,6);
k3(1:2,1:2) = k3t(1:2,1:2)*E*A/L;
k3(1:2,5:6) = k3t(1:2,3:4)*E*A/L;
k3(5:6,1:2) = k3t(3:4,1:2)*E*A/L;
k3(5:6,5:6) = k3t(3:4,3:4)*E*A/L;
k3
%% Global Stiffness Matrix
K = k1+k2+k3
Korig = K
K(:,4) = []; %%manually saying v2=0
K(4,:) = [];
K(:,2) = []; %%manually saying v1=0
K(2,:) = [];
K(:,1) = []; %%manually saying u1=0
K(1,:) = [];
K
%% Solve
syms f3x u2 u3
FVector = K*[u2;u3;u3];
eq1 = P == FVector(1);
eq2 = f3x == FVector(2);
eq3 = f3x == FVector(3);
A = [eq1;eq2;eq3]
B = [f3x;u2;u3]
sol = solve(A,B)
https://preview.redd.it/qegoa3tb9tx71.png?width=686&format=png&auto=webp&s=f81f6a76c454053d6b352a2bd37f4ab38495759a
Hello, I'm having some trouble understanding this solution that was provided.
https://i.imgur.com/5UbFGMW.png
From what I understood, a SLE has a unique solution if each column has a leading 1. In this example, when the SLE is put into REF there's no leading 1.
I also find it confusing that for a unique solution, k can have any value other than 2 or -5. I thought there can only be one answer for k if it's unique.
A company produces software programs and video games.
$50 profit for software
$35 profit for games
Up to 200 programs can be made per week.
Up to 300 games can be made per week.
Max 400 items can be sold per week.
How many of each item should be sold to maximize profit?
Flying against the jetstream, a jet travels 4000 miles in 4 hours. Flying with the jetstream, the same jet travels 11,200 miles in 8 hours. What is the rate of the jet in still air and what is the rate of the jetstream?
I have this question here, and am not quite sure how to go about it. Would greatly appreciate some help, thank you!
Please, a ELI5 version for idiots.
I know that I have a trivial solution where x = y = z = 0. But how do I know by looking at a homogeneous system that I'll either have a trivial solution or infinite many of them?
The professor said that a homogeneous system will always have a solution. Either a trivial solution or infinite many of them.
So I'm in my first year of engineering, learning matrices. I have a test on solving systems of linear equations next monday and I wanna do well. But the problem is I get every single gaussian elimination problem (and cramers rule and inverse matrix method too for that matter, but gaussian elimination is my main struggle). I know how to solve systems of linear equations with the simultaneous equations method just fine and it always comes out right, but when I try solve it with gaussian elimination the answers are always wrong and end up as weird ass fractions like -(37/2). i've tried using the methods shown on my notes and on youtube videos for solving equations using gaussian elimination but it just never ends up working. Personally I find working within matrices very restricting, for example with simulatneous equations I can just perform simple operations to create new equations with different variables eliminated, as you do. But then with gaussian elimation I have to stay strict to the echelon form and staying within the augmented matrix and it just feels like every operation that I think is the correct one just ends up being completely incorrect. You can probably tell by now just from reading this that im not that good at math in general but this is just really pissing me off. Ideally I'd just solve all the questions in the test using simultaneous equations but my lecturer is probably gonna check that we were using matrix operations on our scripts. So please, if anyone has any tips or advice to help me crack solving gaussian elimination and solving systems of linear equations with matrices in general I would really appreciate it.
