The document provides an overview of linear algebra concepts for mathematical geophysics, including: - Definitions of equations, systems of linear algebraic equations, and the Gauss-Jordan reduction method. - Types of systems include unique solution, no solution, and infinitely many solutions. - Einstein summation convention simplifies tensor equations by implicitly summing over repeated indices. - Gaussian elimination uses row operations to put a system of equations in row echelon form and then reduced row echelon form to solve for variables. - Systems can have unique solutions, no solutions, or multiple solutions depending on the relationships between equations and variables.

1. Mathematical Geophysics:
Part 1- Linear Algebra
Contents:
Definition of Equation
System of Linear Algebraic Equation
Gauss-Jordan reduction method
Working Examples
Types of System of Algebraic Equation
according to solution types.
Dr. Amin Khalil

2. What is equation
stands for?
Equation is an analogous to the
weighing balance. It has two sides;
left hand side and right hand side.
Generally known constants are in the
right side, while the unknown
parameters are in the left hand
side. The simplest equation looks
like:
X = 5
Which looks like using a balance to
weigh some vegetables when go
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shopping.

3. Simple balance (equation)
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4. System of Linear
Algebraic Equation
A problem in numerical linear algebra that
requires the solution of n equations in the
unknowns x1, x2,…, xn of the form Ax = b
where A is a square n×n matrix. The solution obtained by computing
the inverse matrix and forming A–1b is less accurate and requires
more arithmetical operations than elimination methods. In
Gaussian elimination multiples of successive equations are added to
all succeeding ones to eliminate the unknowns x1, x2,…, xn–1 in turn.
Properly used, with row interchanges to avoid large multiples, this
leads to a solution that satisfies exactly a system close to the one
given, relative to the machine precision.
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5. Einstein Summation
Convention:
The convention that repeated indices are implicitly summed
over. This can greatly simplify and shorten equations
involving tensors. For example, using Einstein summation,
And
The convention was introduced by Einstein (1916, sec. 5), who
later jested to a friend, "I have made a great discovery in
mathematics; I have suppressed the summation sign every time
that the summation must be made over an index which occurs
twice..." (Kollros 1956; Pais 1982, p. 216).
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6. With Einstein Summation Convention we can
reduce the system of linear algebraic
equation of the form:
A11 x1 +A12 x2 + . . . .+ A1n xn = c1
A21 x1 + A22 x2 + . . . . . +A2n xn = c2
. .
.
.
An1 x1 + An2 x2 . . . . . . .+ Ann xn = cn
Into
or simply Aij xi =cj
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7. Gauss Elimination
Method:
Definition of row reduction makes use of elementary row
The process
operations, and can be divided into two parts. The first part
(sometimes called Forward Elimination) reduces a given system
to row echelon form, from which one can tell whether there are
no solutions, a unique solution, or infinitely many solutions. The
second part (sometimes called back substitution) continues to
use row operations until the solution is found; in other words, it
puts the matrix into reduced row echelon form.
Another point of view, which turns out to be very useful to
analyze the algorithm, is that row reduction produces a matrix
decomposition of the original matrix. The elementary row
operations may be viewed as the multiplication on the left of the
original matrix by elementary matrices. Then the first part of
the algorithm computes an LU decomposition, while the second
part writes the original matrix as the product of a uniquely
determined invertible matrix andKhaliluniquely determined reduced
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row echelon matrix.

8. Gauss elimination applies row
operations of the form:
Type 1: Swap the positions of two
rows.
Type 2: Multiply a row by a
nonzero scalar.
Type 3: Add to one row a scalar
multiple of another.
To produce the echelon form.
What is the echelon form?
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9. Echelon Form:
The leftmost term of a matrix or system of linear
algebraic equation is called the leading term.
Echelon form is obtained through row operation such
that the non-zero leading of row nth occurs to the
right of the leading term of the (n-1) row, i.e if the
leading term of the upper row occurs at column j the
non zero leading term of the lower one will be at j+1
column. The echelon form will make all the elements
under the diagonal of the matrix will be zeros.
Leading term
Diagonal
elements
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10. Examples
For the SLE of the form:
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12. System of linear
algebraic equations
may have:
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13. The unique solution case is illustrated in
the given examples. Now we will see the
case when we have many solutions:
In this example not all the variables have nonzero leading term. Hence in the above example
the z variable cannot be determined uniquely, in
this case it is assigned certain value and the
other unknowns are obtained accordingly.
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14. Describing Solution
set for multi solution
case: the last example the solution set can be
For
described as:
example of system of two
variables free
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15. Another form
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