Introduction Information in science, business, and mathematics is often organized into rows and...
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apter 1 = Systems Linear Equations & Matrices MATH 264 Linear
Introduction Information in science, business, and mathematics is often organized into rows and columns to form rectangular arrays called “matrices” (plural
Introduction Information in science, business, and mathematics
is often organized into rows and columns to form rectangular arrays
called matrices (plural of matrix). Matrices often appear as tables
of numerical data that arise from physical observations, but they
occur in various mathematical contexts as well. Example:
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Linear Equations A line in a 2D plane (or x-y coordinate
system) can be represented as And a line in a 3D plane as The
following are examples of linear equations The following are NOT
linear equations
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A finite set of linear equations are called linear systems (or
systems of linear equations) The variables are our unknowns A
general linear system of m equations in the n unknowns are written
as
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2 & 3 Unknowns in Linear Systems Linear systems in two
unknowns arise in connection with intersections of lines. For
example, consider the linear system in which the graphs of the
equations are lines in the xy-plane. Each solution (x, y) of this
system corresponds to a point of intersection of the lines, so
there are three possibilities
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In general we say that the linear system is consistent if it
has 1 solution No solutions means the linear system is
inconsistent
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Augmented Matrices As the number of equations and unknowns
increase, so does the complexity of solving the system of linear
equations. We can abbreviate the linear system by partitioned
matrixaugmented matrix
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What are Augmented Matrices? An augmented matrix of a system of
linear equations is really just that system without the variables
and signs (+ - x / =) The augmented matrix for
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Elementary Row Operations The basic method for solving a linear
system is to perform appropriate algebraic operations on the system
that do not alter the solution set The 3 common row operations
performed on a matrix: Multiply a row through by a nonzero constant
Switch two distinct rows Add one row to another row
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Example
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Introduction to Elimination In this section we will develop a
systematic procedure for solving systems of linear equations. The
procedure is based on the idea of performing operations on rows of
augmented matrix that simplify it to a form from which the solution
to that system can be found. Almost all of the methods that are
used for solving large systems of equations are based on the simple
laws that we will develop here.
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Reduced Row Echelon Form (RREF) To be RREF a matrix must have:
If a row does not consist entirely of zeros then the first nonzero
number in the row is a 1. If there are any rows that consist
entirely of zeros then they are grouped together @ bottom of matrix
In any 2 successive rows that do not consist entirely of zeros the
leading 1 in the lower row occurs farther to the right than the
leading 1 in the higher row Each column that contains a leading 1
has zeros everywhere else in that column
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A matrix that has the first 3 properties is in ROW ECHELON
FORM. A matrix that has all 4 properties is in REDUCED ROW ECHELON
FORM.
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Example: More on REF | RREF
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Definitions General Solution = A set of parametric equations
from which all solutions can be obtained by assigning numerical
values to the parameters. Applicable only when a linear system has
many solutions. An m X n matrix is a rectangular array of numbers
with m rows and n columns Size and arrangement matters
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Examples of matrices in REF
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Examples of matrices NOT in REF
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The 3 Elementary Row Operations 1. Scale a row by a non-zero
real number 2. Add a non-zero multiple of one row to another 3.
Exchange two rows Example of how to do this is shown on next
slide
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Just keep doing Steps 1 4 for all the other rows
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Just keep doing Steps 1 4 for all the other rows Note: some
steps are missing and the final result is shown below
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Determining solutions from REF of augmented matrix We use
Gaussian elimination to put the augmented matrix of a system of
linear equations in REF (row echelon form). If the augmented matrix
of a system is in row echelon form, we can determine the solutions
using substitution, starting with the last equation and working
upward. This is method of back-substitution
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Possibilities for solutions The system has no solution =
therefore the system is inconsistent The system has a unique
solution = therefore the system is consistent The system has
infinitely many solutions THEOREM: If a REF of the augmented matrix
of a system of equations has a leading 1 in the final column then
the system is inconsistent & has no solutions.
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Example of the Theorem
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More on the Theorem If the row echelon form of the augmented
matrix of a system of linear equations does not have a leading 1 in
the final column then the system has a solution.
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Variables and Parameters Leading variables = the variables
corresponding to the columns which contain a leading 1 Free
variables = variables that are free to take on any value. We
usually assign them an arbitrary constant. Parameters = The
arbitrary constants given to the free variables Different free
variables can have different values so you need to represent their
values with different symbols.
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Gauss Jordan Elimination Gauss-Jordan elimination is an
extension of Gaussian elimination. Once a matrix is in row echelon
form, suitable multiples of lower rows are added to upper rows to
put zeros above each leading 1 starting with the leading 1 in the
lowest row and working upwards.
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Matrices and RREF A matrix is said to be in reduced row echelon
form (RREF) is it is in row echelon form and any column containing
a leading 1 has zeros everywhere else in that column. Gauss-Jordan
Elimination will always result in a matrix in RREF
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Properties of Products If A and B are m x n matrices and x and
y are n x 1 column matrices and k is a real number then:
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For First Column: For Second Column: For Third Column:
Therefore
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