LINEAR EQUATIONS IN LINEAR ALGEBRA 3. Do this for every variable until the system is
Key feature: introduce some of the central concepts clean
of linear algebra in a simple and concrete setting.
ROW OPERATIONS
LINEAR EQUATIONS Replace one equation by the sum of itself
a1x1+a2x2+…+anxn=b and a multiple of another equation
Interchange two equations
Coefficients: a1, a2, …, an Multiply all terms in the equation with a
nonzero constant
Solution: any value of a variable (x1, x2, …, xn) that
makes the specified equation true. Row operations are reversible
Solution set: the set of all variables that makes the
x1 Row equivalent: if a sequence of row operations
equation true, x= x 2 turns one matrix into the other the systems are row
equivalent which also means the two systems have
x3 the same solution set.
LINEAR SYSTEMS ECHELON FORM
Definition: multiple linear equations Definition:
Intersection at one point gives 1 solution All nonzero rows are above any rows of all
Intersection at zero points dives no solution zeros.
Intersection at all points gives infinitely many Each leading entry of a row is in a column to
solutions the right of the leading entry of the row above
it.
Example. All entries in a column below a leading entry
are zeros.
Leading entries may have any nonzero value
Example.
Equivalent linear systems: two linear systems with
the same solution set
MATRICES Reduced echelon form:
Coefficient matrix: the coefficients of each variable The leading entry in each nonzero row is 1
aligned in columns. Each leading 1 is the only nonzero entry in its
column.
Example.
Example.
Augmented matrix: coefficient matrix with an added
column containing the constants from the right side.
Pivot position: a location in the matrix that
corresponds to a leading 1 in the reduces echelon
Example.
form
Pivot column: column that contains a pivot position
Pivot: nonzero number used as needed to create
zero’s via row operations
The size of a matrix is denoted by m x n, where m Forward phase: to create echelon form
stands for the amount of rows and n stands for the Backward phase: to create reduced echelon form
amount of columns
THEOREM 1
SOLVING A LINEAR SYSTEM Each matrix is row equivalent to one and only one
1. Use x1 in the first equation to eliminate x1 in other reduced echelon matrix, the reduced echelon matrix
rows is unique.
2. Use x2 in the second equation to eliminate x2 in
other rows BACK-SUBSTITUTION
,Definition: use a matrix in echelon form (not Example. vector with two entries
reduced) and substitute the values from bottom to
top to get the solution.
Using reduced echelon form is a better strategy! Two vectors are only equal if their corresponding
entries are equal.
VARIABLES
Basic: the variables corresponding to pivot columns
in the matrix Not equal: ,
Free: the other variables (that do not correspond to Equal: vector with its geometric point (a, b)
pivot columns)
Solution set for a consistent system can be Calculation with vectors:
described by solving for the basic variables in Sum of vectors: parallelogram method
terms of the free variables Multiple of vectors: scalar multiples are
‘x is free’ means you can choose any value, vectors trough the origin because original
every different choice determines a solution vectors always start in the origin
of the system
Zero vector: vector whose entries are all zero
PARAMETRIC DESCRIPTION
Definition: a description of the solution set in which LINEAR COMBINATION
free variables act as parameters. Definition: adding up scaled vectors
c1v1+c2v2+…+cnvn
Example.
VECTOR EQUATION
x1v1+x2v2+…+xnvn=b
The vector b is called a linear combination of the
vectors v1, v2, …, vn with weights c1, c2, …, cn.
The weights can be any real number, including 0
A vector equation has the same solution set as the
No parametric description when the system is linear system whose augmented matrix is:
inconsistent. [v1 v2 … vn b], b can be generated as a linear
Many parametric descriptions when the system is combination of the vectors if and only if there exists
consistent with free variables. a solution to the system.
UNIQUENESS AND CONSISTENCY How to solve the question: is this a linear
Consistent system: the system has one or infinitely combination of/ can this be generated as a linear
many solutions combination?
Pivot in every row Determine whether the augmented matrix from the
For Ax=b, b is in the columnspace of A vector equation has a solution and if so for which
weights.
Inconsistent system: the system has no solution
Zero row SPAN
a1
Unique system: the system has only one solution Vectors in R = nx1 matrix = …
n
No free variables an
Pivot in every column
Span {v1, v2, …, vp}: set of all the scaled up
THEOREM 2
combinations of the vectors v1, v2, …, vp.
A linear system is consistent if and only if the
rightmost column of the augmented matrix is not a
For some combinations of vectors:
pivot column. That is if and only if an echelon form of
Span {v1, v2, …, vp} = Rn, so we can represent all the
the augmented matrix has no row of the form:
vectors in Rn with some linear combination of the
vectors in the Span.
[0 … 0 b] with b is nonzero
How to solve the question: is the vector b in the
VECTORS
Span?
Definition: a matrix with only one column Does the vector equation have a solution, or
equivalently does the linear system with augmented
, matrix [v1 v2 … vp] have a solution. has only one vector and the solution set is a
line through the origin
MATRIX EQUATION
Definition: Ax=b PARAMETRIC VECTOR FORM
Solution for homogeneous systems:
Ax is defined only if the number of columns of A x=su+tv
equals the number of entries in x. x= tv
Ax=b has a solution if and only if b is a linear s & t = free variable or number
combination of the columns of A. u & v = vector
How to solve the question: is the equation Ax=b Solutions for nonhomogeneous systems: adding
consistent for all possible b? the vector p to the solutions of Ax=0
Row reduce the augmented matrix and check for a x=p+tv
0=number row, then check what b needs to be for
the system to be consistent is every row already TRANSLATION
consistent then the system is probably consistent for Definition: v is translated by p into v+p
every b. The solution set of Ax=b is a line trough p
parallel to the solution set of Ax=0
The columns of A span Rm: every b in Rm is a Ax=0 is translated into Ax=b by adding p
linear combination of the columns of A
THEOREM 3 THEOREM 6
If A is an mxn matrix with the columns a1, …, an and If the equation Ax=b is consistent for a given b and p
if b is in Rm the matrix equation Ax=b has the same is a solution then the solution set of Ax=b is a set of
solution set as the vector equation: all vectors of the form w=p+vh where vn is any
x1a1+x2a2+…+xnan=b solution of the homogeneous equation Ax=0.
which in turn has the same solution set as the
system of linear equations with augmented matrix: If Ax=b has a solution it is obtained by translating the
[a1 a2 … an] solution set of Ax=0
THEOREM 4 This theorem only applies when the equation Ax=b
Let A be an mxn matrix, for a particular A either all has at least one nonzero solution, when Ax=b has
statements are true or all statements are false: no solution the solution set is empty.
For each b in Rm the equation Ax=b has a
solution LINEAR (IN)DEPENDANCY
Each b in Rm is a linear combination of the Independent: if the vector equation has only the
colums of A trivial solution
The columns of A span Rm No free variables only basic variables
A has a pivot position in every row The vector is independent if the vector is not
the zero vector
This theorem is about a coefficient matrix not an
augmented one Dependent: weights (c1, c2, …, cn) exist such that
c1v1+c2v2+…+cpvp=0
THEOREM 5 Free variables
If A is an mxn matrix, u and v are vectors in Rn and c If one of the vectors is the multiple of the
is a scalar then: other then they are dependent
A(u+v) = Au+Av
A(cu) = c(Au) THEOREM 7
An set S={v1, …, vp} of two or more vectors is linearly
HOMOGENEOUS SYSTEM dependent if and only if at least one vector is a linear
Definition: Ax=0 combination of the others.
Always at least one solution, x=0, called the
trivial solution The theorem does not say that every vector in a
The system has a nontrivial (nonzero) dependent set is a linear combination of the
solution is and only if the equation has at preceding vectors.
least one free variable
Obtain the trivial solution by choosing the THEOREM 8
value 0 for the free variable If a set contains more vectors than there are entries
If the equation has only one free variable it in each vector the set is linearly dependent.
Key feature: introduce some of the central concepts clean
of linear algebra in a simple and concrete setting.
ROW OPERATIONS
LINEAR EQUATIONS Replace one equation by the sum of itself
a1x1+a2x2+…+anxn=b and a multiple of another equation
Interchange two equations
Coefficients: a1, a2, …, an Multiply all terms in the equation with a
nonzero constant
Solution: any value of a variable (x1, x2, …, xn) that
makes the specified equation true. Row operations are reversible
Solution set: the set of all variables that makes the
x1 Row equivalent: if a sequence of row operations
equation true, x= x 2 turns one matrix into the other the systems are row
equivalent which also means the two systems have
x3 the same solution set.
LINEAR SYSTEMS ECHELON FORM
Definition: multiple linear equations Definition:
Intersection at one point gives 1 solution All nonzero rows are above any rows of all
Intersection at zero points dives no solution zeros.
Intersection at all points gives infinitely many Each leading entry of a row is in a column to
solutions the right of the leading entry of the row above
it.
Example. All entries in a column below a leading entry
are zeros.
Leading entries may have any nonzero value
Example.
Equivalent linear systems: two linear systems with
the same solution set
MATRICES Reduced echelon form:
Coefficient matrix: the coefficients of each variable The leading entry in each nonzero row is 1
aligned in columns. Each leading 1 is the only nonzero entry in its
column.
Example.
Example.
Augmented matrix: coefficient matrix with an added
column containing the constants from the right side.
Pivot position: a location in the matrix that
corresponds to a leading 1 in the reduces echelon
Example.
form
Pivot column: column that contains a pivot position
Pivot: nonzero number used as needed to create
zero’s via row operations
The size of a matrix is denoted by m x n, where m Forward phase: to create echelon form
stands for the amount of rows and n stands for the Backward phase: to create reduced echelon form
amount of columns
THEOREM 1
SOLVING A LINEAR SYSTEM Each matrix is row equivalent to one and only one
1. Use x1 in the first equation to eliminate x1 in other reduced echelon matrix, the reduced echelon matrix
rows is unique.
2. Use x2 in the second equation to eliminate x2 in
other rows BACK-SUBSTITUTION
,Definition: use a matrix in echelon form (not Example. vector with two entries
reduced) and substitute the values from bottom to
top to get the solution.
Using reduced echelon form is a better strategy! Two vectors are only equal if their corresponding
entries are equal.
VARIABLES
Basic: the variables corresponding to pivot columns
in the matrix Not equal: ,
Free: the other variables (that do not correspond to Equal: vector with its geometric point (a, b)
pivot columns)
Solution set for a consistent system can be Calculation with vectors:
described by solving for the basic variables in Sum of vectors: parallelogram method
terms of the free variables Multiple of vectors: scalar multiples are
‘x is free’ means you can choose any value, vectors trough the origin because original
every different choice determines a solution vectors always start in the origin
of the system
Zero vector: vector whose entries are all zero
PARAMETRIC DESCRIPTION
Definition: a description of the solution set in which LINEAR COMBINATION
free variables act as parameters. Definition: adding up scaled vectors
c1v1+c2v2+…+cnvn
Example.
VECTOR EQUATION
x1v1+x2v2+…+xnvn=b
The vector b is called a linear combination of the
vectors v1, v2, …, vn with weights c1, c2, …, cn.
The weights can be any real number, including 0
A vector equation has the same solution set as the
No parametric description when the system is linear system whose augmented matrix is:
inconsistent. [v1 v2 … vn b], b can be generated as a linear
Many parametric descriptions when the system is combination of the vectors if and only if there exists
consistent with free variables. a solution to the system.
UNIQUENESS AND CONSISTENCY How to solve the question: is this a linear
Consistent system: the system has one or infinitely combination of/ can this be generated as a linear
many solutions combination?
Pivot in every row Determine whether the augmented matrix from the
For Ax=b, b is in the columnspace of A vector equation has a solution and if so for which
weights.
Inconsistent system: the system has no solution
Zero row SPAN
a1
Unique system: the system has only one solution Vectors in R = nx1 matrix = …
n
No free variables an
Pivot in every column
Span {v1, v2, …, vp}: set of all the scaled up
THEOREM 2
combinations of the vectors v1, v2, …, vp.
A linear system is consistent if and only if the
rightmost column of the augmented matrix is not a
For some combinations of vectors:
pivot column. That is if and only if an echelon form of
Span {v1, v2, …, vp} = Rn, so we can represent all the
the augmented matrix has no row of the form:
vectors in Rn with some linear combination of the
vectors in the Span.
[0 … 0 b] with b is nonzero
How to solve the question: is the vector b in the
VECTORS
Span?
Definition: a matrix with only one column Does the vector equation have a solution, or
equivalently does the linear system with augmented
, matrix [v1 v2 … vp] have a solution. has only one vector and the solution set is a
line through the origin
MATRIX EQUATION
Definition: Ax=b PARAMETRIC VECTOR FORM
Solution for homogeneous systems:
Ax is defined only if the number of columns of A x=su+tv
equals the number of entries in x. x= tv
Ax=b has a solution if and only if b is a linear s & t = free variable or number
combination of the columns of A. u & v = vector
How to solve the question: is the equation Ax=b Solutions for nonhomogeneous systems: adding
consistent for all possible b? the vector p to the solutions of Ax=0
Row reduce the augmented matrix and check for a x=p+tv
0=number row, then check what b needs to be for
the system to be consistent is every row already TRANSLATION
consistent then the system is probably consistent for Definition: v is translated by p into v+p
every b. The solution set of Ax=b is a line trough p
parallel to the solution set of Ax=0
The columns of A span Rm: every b in Rm is a Ax=0 is translated into Ax=b by adding p
linear combination of the columns of A
THEOREM 3 THEOREM 6
If A is an mxn matrix with the columns a1, …, an and If the equation Ax=b is consistent for a given b and p
if b is in Rm the matrix equation Ax=b has the same is a solution then the solution set of Ax=b is a set of
solution set as the vector equation: all vectors of the form w=p+vh where vn is any
x1a1+x2a2+…+xnan=b solution of the homogeneous equation Ax=0.
which in turn has the same solution set as the
system of linear equations with augmented matrix: If Ax=b has a solution it is obtained by translating the
[a1 a2 … an] solution set of Ax=0
THEOREM 4 This theorem only applies when the equation Ax=b
Let A be an mxn matrix, for a particular A either all has at least one nonzero solution, when Ax=b has
statements are true or all statements are false: no solution the solution set is empty.
For each b in Rm the equation Ax=b has a
solution LINEAR (IN)DEPENDANCY
Each b in Rm is a linear combination of the Independent: if the vector equation has only the
colums of A trivial solution
The columns of A span Rm No free variables only basic variables
A has a pivot position in every row The vector is independent if the vector is not
the zero vector
This theorem is about a coefficient matrix not an
augmented one Dependent: weights (c1, c2, …, cn) exist such that
c1v1+c2v2+…+cpvp=0
THEOREM 5 Free variables
If A is an mxn matrix, u and v are vectors in Rn and c If one of the vectors is the multiple of the
is a scalar then: other then they are dependent
A(u+v) = Au+Av
A(cu) = c(Au) THEOREM 7
An set S={v1, …, vp} of two or more vectors is linearly
HOMOGENEOUS SYSTEM dependent if and only if at least one vector is a linear
Definition: Ax=0 combination of the others.
Always at least one solution, x=0, called the
trivial solution The theorem does not say that every vector in a
The system has a nontrivial (nonzero) dependent set is a linear combination of the
solution is and only if the equation has at preceding vectors.
least one free variable
Obtain the trivial solution by choosing the THEOREM 8
value 0 for the free variable If a set contains more vectors than there are entries
If the equation has only one free variable it in each vector the set is linearly dependent.
