Gaussian Elimination is one of the main algorithms for solving systems of linear equations.
Any equation that has an unknown is called a point equation and has only one solution.
Any equation that has two unknowns resolves to a line, and is called a linear equation and has an infinite number of solutions as long as they are on the line.
System(Combination) of linear equations:
Is any combination(above 2) of line equations as long as they intersect together at a certain point and share the same variables.
A good example is:
2y = 4x
3y = 5x
The most intuitive way to solve any system of equation is:
In the first equation, solve for one of the variables in terms of the others.
Substitute this expression into the remaining equations. This yields a system of equations with one fewer equation and one fewer unknown.
Repeat until the system is reduced to a single linear equation.
Solve this equation, and then back-substitute until the entire solution is found.
But there is a better way:
Row Reduction also known as Gaussian elimination):
The linear system is represented as an augmented matrix with the coefficients(multipliers/scalars of the variables) as the row vector and the constants as a column on the right side of the augmented matrix.
Then Row Reduction(EF or RREF) is created by either swapping two rows or multiplying a row by a nonzero scalar or by adding a multiple of one row to another row of the matrix until the lower left-hand corner of the matrix is filled as much as possible with zeros.
This is allowed because the both lines are in the same subspace.
REF:
A matrix is in row echelon form if all rows with at least one nonzero element are above any rows of all zeroes.
Also, the leading coefficient, a.k.a – the pivot, a.k.a the first nonzero element from the left, of a nonzero row is always to the right and below of the leading coefficient of the row above it.
e.g.
3434309
0023110
0000080
when solving a system of linear equations it is better to use REF with all leading elements as 1:
e.g.
013-20
00134
00001
Then the equation is solved by converting the result matrix back to a form of equations with the new reduced coefficients on the left multiplied by the unused variables and solved with the reduced constants on the right.
e.g.
1x+3y-2z =3
00 1y+3z =4
000 1z =1
z=1
y=4-3(z)=1
x=3-3(y)+2(z) = -2
RREF (A.K.A row canonical form):
A matrix is in reduced row echelon form if it is in row echelon form and the leading element in each nonzero row is a 1 and all the elements in the column containing a leading 1 are zeros.
e.g.
1000003
0010301
0000010
Then the equation is solved by converting the result matrix back to a form of equations with the new reduced coefficients on the left multiplied by the unused variables and solved with the reduced constants on the right.
In reduced row echelon form you can just read off the answer because all variables are zeros in above and below rows:
e.g.
1x 00000 =3
001y+0+3z0=1
000001v =0
x=3
y+3z=1
v=0