augmented matrix - ANSA matrix made up of a coefficient matrix for a linear system and
one or more columns to the right.
backward phase - ANSThe last part of the algorithm that reduces a matrix in echelon
form to a reduced echelon form.
basic variable - ANSA variable in a linear system that corresponds to a pivot column in
the coefficient matrix.
codomain - ANSThe set Rm that contains the range of T. In general, if T maps a vector
space V into a vector space W, then W is called the domain of T.
coefficient matrix - ANSA matrix whose entries are the coefficients of a system of linear
equations.
coefficients - ANS
column vector (vector) - ANSA matrix with only one column, or a single column of a
matrix that has several columns.
consistent linear system - ANSA system with at least one solution.
contraction - ANS
dilation - ANSA mapping x > rx for some scalar, with 1 < r.
domain - ANSThe set of all vectors x for which T(x) is defined.
echelon matrix - ANSA rectangular matrix that has three properties: (1) All nonzero rows
above any row of all zeros. (2) Each leading entry of a row is in a column to the right of
the leading entry of the row above it. (3) All entries in a column below a leading entry
are zero.
equivalent linear systems - ANSLinear systems with the same solution set.
flop - ANSOne arithmetic operation (+, -, *, /) on two real floating point numbers.
forward phase - ANSThe first part of the algorithm that reduces a matrix to echelon
form.
free variable - ANSAny variable in a linear system that is not a basic variable.
homogeneous equation - ANSAn equation of the form Ax = 0, possibly written as a
vector equation or as a system of linear equations.
image - ANSThe vector T(x) assigned to x by T.
,inconsistent linear system - ANSA system with no solutions.
leading entry - ANSThe leftmost nonzero entry in a row of a matrix.
linear combination - ANSA sum of scalar multiples of vectors.
linear dependence relation - ANSA homogeneous vector equation where the weights
are all specified and at least one weight is nonzero.
linear equation - ANSAn equation that can be written in the form a1x1 + a2x2 + ... +
anxn = b, where b and the coefficients a1, ..., an are real or complex numbers.
m x n matrix - ANSA matrix with m rows and n columns.
matrix - ANSA rectangular array of numbers.
matrix equation - ANSAn equation that involves at least one matrix; for instance, Ax = b.
nontrivial solution - ANSA nonzero solution of a homogeneous equation or system of
homogeneous equations.
parametric vector equation - ANS
parametric vector form - ANS
pivot - ANSA nonzero number that either is used in a pivot position to create zeros
through row operations or is changed into a leading 1, which in turn is used to create
zeros.
pivot column - ANSA column that contains a pivot position.
pivot position - ANSA position in a matrix A that corresponds to a leading entry in an
echelon form of A.
product Ax - ANSThe linear combination of the columns of A using the corresponding
entries in x as weights.
range - ANSThe set off all vectors of the form T(x) for some x in the domain of T.
reduced echelon matrix - ANSA rectangular matrix in echelon form that has these
additional properties: The leading entry in each nonzero row is 1, and the leading 1 is
the only nonzero entry in its column.
row equivalent - ANSTwo matrices for which there exists a finite sequence of row
operations that transforms one matrix into the other.
, row reduced - ANS
shear transformation - ANS
size - ANSTwo numbers, written in the form m x n, that specify the number of rows (m)
and columns (n) in the matrix.
solution - ANSA list (s1, s2, ..., sn) of numbers that makes each equation in the system
a true statement when the values s1, ..., sn are substituted for x1, ..., xn, respectively.
solution set - ANSThe set of all possible solutions of a linear system.
Span{v1, ..., vp} - ANSThe set of all linear combinations of v1, ..., vp.
subset - ANS
system of linear equations (linear system) - ANSA collection of one or more linear
equations involving the same set of variables, say, x1, ... , xn.
transformation (mapping) - ANSA rule that assigns each vector x in Rn a unique vector
T(x) in Rm.
trivial solution - ANSThe solution x = 0 of a homogeneous equation Ax = 0
weights - ANSThe scalars used in a linear combination.
zero vector - ANSThe unique vector, denoted by a bold 0, such that u + 0 = u for all u.
augmented matrix - ANSA matrix made up of a coefficient matrix for a linear system and
one or more columns to the right.
backward phase - ANSThe last part of the algorithm that reduces a matrix in echelon
form to a reduced echelon form.
basic variable - ANSA variable in a linear system that corresponds to a pivot column in
the coefficient matrix.
codomain - ANSThe set Rm that contains the range of T. In general, if T maps a vector
space V into a vector space W, then W is called the domain of T.
coefficient matrix - ANSA matrix whose entries are the coefficients of a system of linear
equations.
coefficients - ANS
one or more columns to the right.
backward phase - ANSThe last part of the algorithm that reduces a matrix in echelon
form to a reduced echelon form.
basic variable - ANSA variable in a linear system that corresponds to a pivot column in
the coefficient matrix.
codomain - ANSThe set Rm that contains the range of T. In general, if T maps a vector
space V into a vector space W, then W is called the domain of T.
coefficient matrix - ANSA matrix whose entries are the coefficients of a system of linear
equations.
coefficients - ANS
column vector (vector) - ANSA matrix with only one column, or a single column of a
matrix that has several columns.
consistent linear system - ANSA system with at least one solution.
contraction - ANS
dilation - ANSA mapping x > rx for some scalar, with 1 < r.
domain - ANSThe set of all vectors x for which T(x) is defined.
echelon matrix - ANSA rectangular matrix that has three properties: (1) All nonzero rows
above any row of all zeros. (2) Each leading entry of a row is in a column to the right of
the leading entry of the row above it. (3) All entries in a column below a leading entry
are zero.
equivalent linear systems - ANSLinear systems with the same solution set.
flop - ANSOne arithmetic operation (+, -, *, /) on two real floating point numbers.
forward phase - ANSThe first part of the algorithm that reduces a matrix to echelon
form.
free variable - ANSAny variable in a linear system that is not a basic variable.
homogeneous equation - ANSAn equation of the form Ax = 0, possibly written as a
vector equation or as a system of linear equations.
image - ANSThe vector T(x) assigned to x by T.
,inconsistent linear system - ANSA system with no solutions.
leading entry - ANSThe leftmost nonzero entry in a row of a matrix.
linear combination - ANSA sum of scalar multiples of vectors.
linear dependence relation - ANSA homogeneous vector equation where the weights
are all specified and at least one weight is nonzero.
linear equation - ANSAn equation that can be written in the form a1x1 + a2x2 + ... +
anxn = b, where b and the coefficients a1, ..., an are real or complex numbers.
m x n matrix - ANSA matrix with m rows and n columns.
matrix - ANSA rectangular array of numbers.
matrix equation - ANSAn equation that involves at least one matrix; for instance, Ax = b.
nontrivial solution - ANSA nonzero solution of a homogeneous equation or system of
homogeneous equations.
parametric vector equation - ANS
parametric vector form - ANS
pivot - ANSA nonzero number that either is used in a pivot position to create zeros
through row operations or is changed into a leading 1, which in turn is used to create
zeros.
pivot column - ANSA column that contains a pivot position.
pivot position - ANSA position in a matrix A that corresponds to a leading entry in an
echelon form of A.
product Ax - ANSThe linear combination of the columns of A using the corresponding
entries in x as weights.
range - ANSThe set off all vectors of the form T(x) for some x in the domain of T.
reduced echelon matrix - ANSA rectangular matrix in echelon form that has these
additional properties: The leading entry in each nonzero row is 1, and the leading 1 is
the only nonzero entry in its column.
row equivalent - ANSTwo matrices for which there exists a finite sequence of row
operations that transforms one matrix into the other.
, row reduced - ANS
shear transformation - ANS
size - ANSTwo numbers, written in the form m x n, that specify the number of rows (m)
and columns (n) in the matrix.
solution - ANSA list (s1, s2, ..., sn) of numbers that makes each equation in the system
a true statement when the values s1, ..., sn are substituted for x1, ..., xn, respectively.
solution set - ANSThe set of all possible solutions of a linear system.
Span{v1, ..., vp} - ANSThe set of all linear combinations of v1, ..., vp.
subset - ANS
system of linear equations (linear system) - ANSA collection of one or more linear
equations involving the same set of variables, say, x1, ... , xn.
transformation (mapping) - ANSA rule that assigns each vector x in Rn a unique vector
T(x) in Rm.
trivial solution - ANSThe solution x = 0 of a homogeneous equation Ax = 0
weights - ANSThe scalars used in a linear combination.
zero vector - ANSThe unique vector, denoted by a bold 0, such that u + 0 = u for all u.
augmented matrix - ANSA matrix made up of a coefficient matrix for a linear system and
one or more columns to the right.
backward phase - ANSThe last part of the algorithm that reduces a matrix in echelon
form to a reduced echelon form.
basic variable - ANSA variable in a linear system that corresponds to a pivot column in
the coefficient matrix.
codomain - ANSThe set Rm that contains the range of T. In general, if T maps a vector
space V into a vector space W, then W is called the domain of T.
coefficient matrix - ANSA matrix whose entries are the coefficients of a system of linear
equations.
coefficients - ANS
