**Unformatted text preview: **elon form of a matrix is unique.
The columns containing leading 1’s are called pivot columns.
The variables corresponding to pivot columns are called
leading variables.
The the variables that are not leading variables are called free
variables. Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Examples Example
Use Gauss-Jordan elimination to solve the linear systems.
1 3x1 + 2x2 − x3 = 4
x1 − 2x2 + 2x3 = 1
11x1 + 2x2 + 4x3 = 14
2 x1 + x2 + x3 + x4 + x5 = 5
x1 + x2 + 3x3 + x4 + 3x5 = 1
x1 + x2 + 2x3 + x4 + 3x5 = 5
(This system is underdetermined) Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Homogeneous Linear Systems
A homogeneous system of linear equations is a system of
linear equations with zero constant term, so its augmented
matrix has the form: a11 a12 · · · a1n 0 a21 a22 · · · a2n 0 (6) .
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am 1 am 2 · · · amn 0 Every homogeneous system of linear equations is consistent
because the trivial solution x1 = x2 = · · · = xn = 0 is a
solution of the system.
If there are other solutions, they are called nontrivial solutions. Outline Linear Systems Matrices Gaussian Elimination Homogeneous Systems Homogeneous Linear Systems There are two possibilities for the solutions of a homogeneous
system of linear equations:
There is only the trivial solution.
There is the trivial solution and inﬁnitely many nontrivial
solutions. Proposition
A homogeneous system of linear equations having more unknowns
than equations has inﬁnitely many nontrivial solutions....

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