For questions on or related to the technique of Gaussian elimination, used in solving systems of linear equations.

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0answers
24 views

Gaussian Elimination vs matrix inversion [on hold]

Why Gaussian Elimination is better than matrix inversion in therms of FLOPS? Also how LU decomposition improves the shifted inverse power method?
2
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1answer
54 views

Gaussian elimination algorithm performance

I am developing the quadratic sieve algorithm and I reached a new bottle neck: The matrix processing. I been reading quit a lot about this topic and I found many solutions Gaussian elimination: ...
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0answers
11 views

Solving a large set of equations with Newton Raphson and Gauss Jordan Elimination in VBA Excel (6x6)

What I tried to do is to solve a system of 6 equations with VBA excel. This system counts some nonlinear equations which requires Newton Raphsons method to solve and Gauss Jordan Elimination to invert ...
0
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1answer
20 views

Given a set of (0,1)-vectors, find a subset which adds to the zero vector mod 2

I am reading Quadratic sieve in wiki and it present the next problem: Given a set of (0,1)-vectors, find a subset which adds to the zero vector mod 2 My question is simple: How to solve it? ...
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1answer
19 views

Finding solutions using Gauß-Jordan-Algorithm

I have matrix A = \begin{bmatrix}1&2&3&4&5\\2&4&3&5&4\\3&6&5&8&7\end{bmatrix} and\vec b = \begin{bmatrix}1\\2\\3\end{bmatrix} I expanded the matrix ...
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24 views

how to solve this equation using gaussian method?

$2x +z+4v= -5$ $2x -3y-z+2w+3v = 4$ $4x-4y-z+4w+11v=4$ $2x-5y-2z+2w-v=9$ here are my steps: interchange R1<>R3 R1 > R1/4 R2>R2 + (-2R1) R4>R4+(-2R1) R2> R2/-1 R3>R3 + (-2R2) R4>R4 +3R2 R3>R3/-1 I ...
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1answer
39 views

Solving matrix using Gaussian elimination and a parameter

$\begin{bmatrix} x_{1} & 2x_{2} & & & ax_{5} & x_{6} & = & -2 \\ -x_{1} & -2x_{2} & & & (-1-a)x_{5} ...
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2answers
48 views

how to find null space basis directly by matrix calculation

The problem of finding the basis for the null space of an $m \times n$ matrix $A$ is a well-known problem of linear algebra. We solve $Ax=0$ by Gaussian elimination. Either the solution is unique and ...
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1answer
75 views

Minimize the sum of solution of linear equation

Let x(i,j) be a variable. All variables and constants can only have value of 0 or 1. Also, sum of two variables x(i,j) and x(k,l) is equal to (x(i,j)+x(k,l)) % 2 ...
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0answers
27 views

Unexpected rank of a matrix

I have a sparse matrix with 100 rows and, when I do a Gauss decomposition, I get a matrix with 90 rows. But if I remove tha last row of the first matrix, now with 99 rows, and I do a Gauss ...
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0answers
45 views

How to solve linear system of equation over a finite field

How can we solve linear system of equation over a finite field? I just found out about finite fields and i am having a hard time understanding solutions given on net. I know its equivalent to ...
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2answers
42 views

How do I choose a free variable?

I have a question regarding the Gaussian method for solving linear equations. I had to solve 2 equations with 3 unknowns and naturally with the elimination process I had 2 variables left. I thought ...
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2answers
49 views

How can we solve the linear system? [closed]

How can we solve the following linear system ? $$ax+by=0 \\ cx+dy\neq 0$$
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1answer
21 views

Gaussian elimination in linear equations

So I have problems with solving this question, can someone please help me out with this one? I really don't need a direct solution I just need to know how should I do it, or if u are generous enough ...
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1answer
65 views

Complete Pivoting VS Partial Pivoting in Gauss Elimination

I have a hard time understanding that when and under what conditions we can use Gauss elimination with complete pivoting, and when with partial pivoting, and when with no pivoting? (I mean what is the ...
0
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0answers
32 views

What does the size of a matrix tell about its dimensions?

I am having trouble understanding a lot of terminology here. When inspecting a matrix columns and rows, what does it indicate -- dimension of the vectors maybe, or anything else? Following that, what ...
0
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0answers
32 views

Finding a linearly independent subset with the same span as a given linearly dependent set of vectors.

Given the following linearly dependent set of vectors in $\mathbb{R^3}$: $$\{(1,1,1)^T, (2,3,1)^T, (4,5,3)^T, (1,2,0)^T\}$$ how would I find a linearly independent subset with the same span? My ...
6
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4answers
276 views

What is an intuitive explanation to why elimination retains solution to the original system of equations?

I've studied linear algebra before, however, I wanted to come back to the foundations and understand it again from the beginning. I was looking the following inoffensive linear equations: $$ x - 2y ...
1
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2answers
42 views

Explanation of Gauss-Jordan elimination method.

I know how to solve the system of linear equations, how to find inverse of matrix etc. by the Gauss-Jordan method. But I want to understand why this method works (in cases of inverse matrix ...
0
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1answer
23 views

Basic Gaussian elimination

I just finished a course about systems of linear équations and I'm trying to do some exercises. Here is the system to solve using Gaussian elimination: $$ \begin{cases} -5x-2y+z=a \\ -4x-4y-4z=b \\ ...
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2answers
82 views

Inverting a matrix in $\mathbb{Z}/n\mathbb{Z}$.

So in my Linear Algebra course I was shown that we cannot directly use row reduction to invert a matrix over a commutative ring in general because the algorithm requires elements to be invertible ...
2
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2answers
81 views

Solving a system of linear equations with parameters

I've been given the following system of equations: $$ (4-\lambda)x_1-2x_2-x_3=1\\ -2x_1+(1-\lambda)x_2+2x_3=2\\ -x_1+2x_2+(4-\lambda)x_3=1 $$ The resulting coefficient matrix would be: $$ ...
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1answer
46 views

In Gauss-Jordan elimination, what does it mean to 'restore' a row?

As seen here I am familiar with the elementary row operations but I've never heard an step worded in this way. Thanks!
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1answer
48 views

Determine if the linear systems $A \vec x = \vec 0$ and $B \vec x = \vec 0$ are equivalent.

For one of my homework assignments, the question posed is as follows: Determine if the linear systems $A \vec x = \vec 0$ and $B \vec x = \vec 0$ are equivalent where: see matrices To solve, I ...
2
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1answer
38 views

Row & Column Operation to Determine Rank

While evaluating the rank of a matrix is it permissible to apply row and column operations simultaneously on a single matrix? Most of the books that I discussed use either row or column operation (but ...
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1answer
34 views

Matrix question, all zeros with constant

What does it mean if the bottom row of a matrix is all zeros followed by a constant? Example Solve the given system of equations using either Gaussian or Gauss-Jordan elimination. x - y + z = 0 -x + ...
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2answers
39 views

Show that n(n+1)/2 multiplications are required

$a_{11}x_1$+$a_{12}x_2$+$a_{13}x_3$+ ...+ $a_{1,n-1}x_{n-1}$+$a_{1n}x_n$ =$b_1$ $a_{22}x_2$+$a_{23}x_3$+ ...+ $a_{2,n-1}x_{n-1}$+$a_{2n}x_n$ =$b_2$ $a_{33}x_3$+ ...+ $a_{3,n-1}x_{n-1}$+$a_{3n}x_n$ ...
0
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0answers
52 views

Finding matrix inverse using Gauss method

I have been trying to find the inverse of a matrix using Gauss method and I want to know suppose what happens if I don't get the "1" in reduced matrix on the left? Does it mean that the inverse ...
0
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0answers
46 views

numerical stability: LU decomposition

I'm trying to evaluate the numerical stability of LU decomposition. I implemented code in java to calculate the inverse matrix with LU. I made 3 attemps. a) mantissa 4 b) mantissa 6 c) maschine ...
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1answer
34 views

Use Gaussian elimination to convert matrix A to row echelon form R.

Can someone please help me with this question if you can? I have done the ERO's, but I did 5 instead of the 3 that it is asking for and I cannot seem to get it down to 3. I'm not sure if I am reading ...
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0answers
28 views

Lower Triangular Form

Suppose that A is an nxn matrix of real numbers such that each entry satisfies |aij| <= 1. If Gaussian elimination with partial pivoting is used to reduce the matrix to lower triangular form, show ...
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0answers
74 views

Gaussian Elimination on Block Matrix

I have a large matrix, with possibly over 100,000,000 elements in it and I want to solve it quickly. I want to take advantage of the fact that the matrix is partitioned into a small number of unique ...
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0answers
14 views

Linear algebra: relationship between vectors and lines/planes/hyperplanes

In my linear algebra class and in every online source I can find, they seem to want to introduce matrices and row reduction as the intersection of lines/planes/hyperplanes (depending on dimension). ...
2
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1answer
42 views

Basin of attraction (Eigenvectors)

First of all, the transition matrix is given as: $$ M= \begin{matrix} \frac 45 & 0 & 0 \\ 0 & \frac 65 & 0 \\ 0 & 0 & 1 \\ \end{matrix} $$ ...
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3answers
105 views

Gaussian elimination involving parameters

The problem is :Solve the given system of equations involving the parameter a : $$x+y+az=1\\ x+ay+z=a\\ ax+y+z=a^2\\ ax+ay+az=a^3 .$$ I tried to solve this using the Gaussian method but I'm stuck ...
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4answers
55 views

Consider the following system and find the values of b for which the system has a solution

So I have this system: $$\left\{\begin{array}{c} x_1 &− x_2 &+ 2x_3 &= 2 \\ x_1 &+ 2x_2 &− x_3 &= 2 \\ x_1 &+ x_2 & &= 2 \\ x_1 & & +x_3 ...
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1answer
17 views

Method to find the set S of reals $λ$ such as $rg($M-I3)<3 given a matrix

Considering the endomorphism $f$ of $R^3$ of \begin{bmatrix} -3 & 5 & -5\\ -4 & 6 & -5\\ -4& 4 &-3 \end{bmatrix} relatively of the canonical base bc of $R^3$ find the ...
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2answers
59 views

Using linear algebra to find constants in the equation of a circle which passes through given points.

Find constants $a ,\ b ,\ c \ $ such that the equation of the circle, $x^2+y^2+ax+by=c$, contains the points $(6,8)$, $(8,4)$, and $(3,9)$. Use the points to create a system: ...
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1answer
22 views

Count of solutions to matrix equations

Given these modular equations: $$a_{1,1} x_1 + a_{1,2} x_2 + \cdots + a_{1,n} x_n = b_1 \bmod p $$ $$a_{2,1} x_1 + a_{2,2} x_2 + \cdots + a_{2,n} x_n = b_2 \bmod p $$ $$\vdots$$ $$a_{m,1} x_1 + ...
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1answer
85 views

Show that if the leading principal minors of a nonsingular $n\times n$ matrix $A$ are all nonzero then the matrix $A$ has $LU$ factorization

I am stucked at this problem: Prove by induction that if the leading principal minors of an $n\times n$ nonsingular matrix $A$ are all nonzero then the matrix $A$ has $LU$ factorization. (The ...
2
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0answers
432 views

Number of Arithmetic Operations in Gaussian-elimination/Gauss-Jordan Hybrid Method for Solving Linear Systems

I am stucked at this problem from the book Numerical Analysis 8-th Edition (Burden) (Exercise 6.1.16) : Consider the following Gaussian-elimination/Gauss-Jordan hybrid method for solving linear ...
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1answer
42 views

How many solutions depending on the parameter (augmented matrix?)

I have to find how many solutions have got the following equations, depending on p parameter? $ \begin{bmatrix} 5 & p & 5 \\ 1 & 1 & 1 \\ p & p & 2 \end{bmatrix} $ $ ...
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1answer
22 views

Gaussian elimination and pivots

I'm having trouble answering the question posed below ("Question: Which x on the left..."). It seems like there's only 2 options as the first two along the diagonal are already boldfaced. I'm not ...
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2answers
39 views

matrix elementary column operations

Till now i was using the elementry row operations to do the gaussian elemination or to calculate the inverse of a matrix. As i started learning the Laplace's transformation to calculate the ...
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2answers
42 views

Solution of System of linear equations

I have three equations: $$ \begin{cases} 4y + z = 2\\ 2x + 6y - 2z = 3\\ 4x + 8y - 5z = 4 \end{cases} $$ Applying Gauss elimination I get: $$ \left[ \begin{array}{ccc|c} ...
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0answers
45 views

Can I go from the LU factorization of a symmetric matrix to its Cholesky factorization, without starting over?

I mistakenly computed the LU factorization and then realized that the question is asking for a Cholesky factorization, i.e., finding a lower triangular matrix L such that the symmetric matrix A has ...
2
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3answers
107 views

Why doesn't Gaussian elimination change the solution set?

Of course, Gaussian elimination is safe to use as proven by the countless systems I've solved with it while practising my linear algebra (which is I must add very basic and low-level), but when Jim ...
0
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1answer
99 views

Gaussian Elimination General Solution

Find the general solution of the following system of equations: Using Gaussian Elimination I was able to get the following solutions for these equations: x = 2 y = 1 z = 0 However, this is not ...
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1answer
73 views

gauss jordan matrix involving parameter $k$

Could anyone help me in solving this matrix? $$\left[\begin{array}{ccc|c} k+2& k-1& k& 2\\ 0& k+2& 2& 0\\ 0& 0& k^2+k-2& k+2 ...
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0answers
39 views

Incomplete Gauss Jordan elimination: what have I left out

I wrote some C++ to implement a Gauss-Jordan elimination: it's all a bit rough and ready (I wanted to get a fix on the algorithm rather than write a piece of "production" code) but you can see it ...