Solving Systems of Linear Equations

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Problem solving in preparation for the quiz
Linear Algebra Concepts
Vector Spaces, Linear Independence
Orthogonal Vectors, Bases
Matrices

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Solving systems of linear equations (Chapter 9)
Graphical methods

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Gauss elimination

Solving systems of linear equations

Matrices provide a concise notation for representing and solving
simultaneous linear equations:

Solving systems of linear equations in Matlab

Two ways to solve systems of linear algebraic equations [A]{x}={b}:
Left- division
x = A\b
Matrix inversion
x = inv(A)*b

Matrix inversion only works for square, non-singular systems; it is
less efficient than left-division.

Solving graphically systems of linear equations

For small sets of simultaneous equations, graphing them and
determining the location of the intersection of the straight line
representing each equation provides a solution.

There is no guarantee that one can find the solution of system of linear
equations:

a) No solution exists
b) Infinite solutions exist
c) System is ill-conditioned

Determinant of the square matrix

Here the coefficient of is called the co factor of A
A cofactor is a polynomial in the remaining rows of A and can be
described as the partial derivative of A. The cofactor polynomial
contains only entries from an (n-1)x (n-1) matrix called a “minor”
obtained from A by eliminating row i and column j.

Determinants of several matrices

Determinants for 1x1, 2x2, 3x3 matrices are

Determinants for square matrices larger than 3 x 3 are more complicated.

Properties of the determinants

If we permute two rows of the rectangular matrix A then the sign of
the determinant det(A) changes.
The determinant of the transpose of a matrix A is equal to the
determinant of the original matrix.
If two rows of A are identical then |A|=0

Cramer’ s Rule

Consider the system of linear equations:

[A]{x}={b}

Each unknown in a system of linear algebraic equations may be
expressed as a fraction of two determinants with denominator D and
with the numerator obtained from D by replacing the column of
coefficients of the unknown in question by the vector b consisting of
constants .

Example of the Cramer’s Rule

Find in the fol lowing system of equations:

Find the determinant D

Find determinant by replacing D’s second column with b

Divide