MA 322 Flashcards
linear equation
in the form
a1*x1 + a2*x2 … an*xn = b
a is constant; x is variable
system of linear equations
(linear system)
collection of one or more
linear equations with
same variables
solution set of
system of equations
list of number that
each make equation true
equivalent linear systems
have same solution set
consistent
has one or infinitely many solutions
inconsistent
has no solutions
coefficent matrix
matrix of coeffiecnts of linear system
augmented matrix
coefficent matrix with constants from
right side of equation
row equivalent
two matrices are row equivalent if
there is a sequence of elementary
row opertations that transforms one into
another
existence
is the matrix consistent?
uniqueness
is there only one solution?
leading entry
left-most non-zero entry in a row
echelon form
(row echelon form)
- All non-zero rows are above any rows of all zeroes
- Each leading entry of a row is in a column to the right of the leading entry of the row above it
- All entries in a column below a leading entry are zero
reduced echelon form
(reduced row echelon form)
In echelon form:
- the entry in each row is one
- each leading entry is the only non-zero entry in its column
pivot position
location that corresponds to a leading one
in the reduced echelon form
pivot column
column that contains a pivot position
pivot
non-zero number in a pivot position
that is needed to create zeroes
in row operations
basic variables
a variable in a linear system that
corresponds to a pivot column
in the coeffient matrix
free variables
any variable that is not a
basic varible
parametric description
uses free varibles as parameters
in form:
x1 = a +x3
x2 = b -x3
x3 is free
x4 = c
vector
(column vector)
matrix with only one column
linear combination
with vectors v1,v2, … vpinRn
and weights (scalars) c1, c2, … cp
in form:
y = c1*v1 + c2*v2… + cp*vp
where y is the linear combination
span
for v1, … vp in Rn
the set of all linear combinations
of v1 … vp is denoted by Span{v1…vp} and is called
the subset of Rn spanned by
v1… vp
Span{u,v} in R3
the set of all scalar multiples of v
(set of points on the line through v and 0)
Span{u,v} in R3
the plane that contains u, v, and 0
if A is an m X n matrix
with columns a1… an and if x is in Rn
and the # of columns of A = # of entries in x
then the product of A and x is:
the linear combination of the columns of A using the
corresponding entries in x as weights
Ax = b
matrix equation
Ax = b
in form
the equation Ax = b
has a solution
if and only if
b is a linear combination
of the columns of A
For A is an m X n matrix
the following statements are either
all true or all false
(about coefficent matrix)
a. ) for each b in Rm the equation Ax = b has a solution
b. ) each b in Rm is a linear combination of the colmns of A
c. ) the cloumns of A span Rm
d. ) A has a pivot position in every row
identity matrix
(I or In)
a square matrix with ones on the diagonal
and zeroes elsewhere
(In * x = x for all x in Rn)
if A is an m X n matrix
u and v are vectors in Rn
and c is a scalar
a. ) A(u + v) = Au + Av
b. ) A(cu) = c(Au)
homogeneous
can be written in form Ax = 0
always has one solution
x = 0
(trival solution)
the homogenous equation
Ax = 0 has a non-trival solution
if and only if
the equation has at least one free variable
in the form
a1*x1 + a2*x2 … an*xn = b
a is constant; x is variable
linear equation
collection of one or more
linear equations with
same variables
system of linear equations
(linear system)
list of number that
each make equation true
solution set of
system of equations
have same solution set
equivalent linear systems
has one or infinitely many solutions
consistent
has no solutions
inconsistent
matrix of coeffiecnts of linear system
coefficent matrix
coefficent matrix with constants from
right side of equation
augmented matrix
is the matrix consistent?
existence
is there only one solution?
uniqueness
left-most non-zero entry in a row
leading entry
- All non-zero rows are above any rows of all zeroes
- Each leading entry of a row is in a column to the right of the leading entry of the row above it
- All entries in a column below a leading entry are zero
echelon form
(row echelon form)
In echelon form:
- the entry in each row is one
- each leading entry is the only non-zero entry in its column
reduced echelon form
(reduced row echelon form)
location that corresponds to a leading one
in the reduced echelon form
pivot position
column that contains a pivot position
pivot column
non-zero number in a pivot position
that is needed to create zeroes
in row operations
pivot
a variable in a linear system that
corresponds to a pivot column
in the coeffient matrix
basic variables
any variable that is not a
basic varible
free variables
uses free varibles as parameters
in form:
x1 = a +x3
x2 = b -x3
x3 is free
x4 = c
parametric description
matrix with only one column
vector
(column vector)
in form:
y = c1*v1 + c2*v2… + cp*vp
where y is the linear combination
linear combination
with vectors v1,v2, … vpinRn
and weights (scalars) c1, c2, … cp
the set of all linear combinations
of v1 … vp is denoted by Span{v1…vp} and is called
the subset of Rn spanned by
v1… vp
span
for v1, … vp in Rn
the set of all scalar multiples of v
(set of points on the line through v and 0)
Span{u,v} in R3
the plane that contains u, v, and 0
Span{u,v} in R3
the linear combination of the columns of A using the
corresponding entries in x as weights
Ax = b
if A is an m X n matrix
with columns a1… an and if x is in Rn
and the # of columns of A = # of entries in x
then the product of A and x is:
Ax = b
in form
matrix equation
if and only if
b is a linear combination
of the columns of A
the equation Ax = b
has a solution
a. ) for each b in Rm the equation Ax = b has a solution
b. ) each b in Rm is a linear combination of the colmns of A
c. ) the cloumns of A span Rm
d. ) A has a pivot position in every row
For A is an m X n matrix
the following statements are either
all true or all false
(about coefficent matrix)
a square matrix with ones on the diagonal
and zeroes elsewhere
(In * x = x for all x in Rn)
identity matrix
(I or In)
a. ) A(u + v) = Au + Av
b. ) A(cu) = c(Au)
if A is an m X n matrix
u and v are vectors in Rn
and c is a scalar
can be written in form Ax = 0
always has one solution
x = 0
(trival solution)
homogeneous
if and only if
the equation has at least one free variable
the homogenous equation
Ax = 0 has a non-trival solution
