Vocab Flashcards
linear equation
an equation that can be written in the form a1x1 + … + anxn = b with b and the coefficients a1…an are real/complex numbers
system of linear equations/ linear system
collection of one or more linear equations involving the same variables
linear systems are equivalent if
they have the same solution set
two matrices are row equivalent
if there is a sequence of elementary row operations that transforms one matrix to another
leading entry
the leftmost nonzero number in a nonzero row
basic variables
variables that correspond to pivot columns
vector / column vector
matrix with only one column
two vectors are equal if
their corresponding entries are equal
linear combination
y = c1v1 + … + cnvn given vectors v1,…,vn, and scalars c1,…,cn
homogeneous
Ax=0
trivial solution
- when x=0
- every homogeneous equation has the trivial solution
nontrivial solution
- x is nonzero
- occurs when there is a free variable
linearly independent
set of vectors {v1, … , vp} with the vector equation x1v1+ … + xpvp = 0 with only the trivial solution
linearly dependent
there exists weights c1, …, cp not all zeros such that c1v1 + …+ cpvp = 0
transformation / function / mapping
a rule that assigns each vector x in Rn a vector T(x) in Rm
Rn
domain
Rm
codomain
T(x)
image
range
set of all T(x)
a transformation is linear if
T(0) = 0 T(u+v) = T(u) + T(v) T(cu) = cT(u)
onto
each b in Rm is the image of at least one x in Rn
one-to-one
each b in Rm is the image of at most one x in Rn
two matrices are equal if
they have the same size (same number of rows and columns)
nonsingular matrix
invertible matrix
elementary matrices
obtained by performing one elementary row operation on the identity matrix
subspace
- includes the zero vector
- is closed under vector addition
- is closed under scalar multiplication
isomorphism from V onto W
one-to-one transformation from a vector space V onto vector space W
dim V
number of vectors in a basis for V
Row A (row space)
set of all linear combinations of the row vectors
rank
dimension of column space of A
nullity
dimension of null space
probability vector
vector with nonnegative entries that add up to 1
stochastic matrix
square matrix whose columns are probability vectors
steady-state vector / equilibrium vector
probability vector q such that Pq=q
stochastic matrix is regular if
some matrix power P^k contains only positive entries
eigenvector
NONZERO vector x such that Ax=(lambda)x
eigenspace
set of all solutions of (A-(lambda)I)x=0
characteristic equation
det(A-(lambda)I)=0
multiplicity
number of times it appears as a root of the characteristic equation
two matrices A and B are similar if
there is an invertible matrix such that A=PBP^-1
diagonalizable
matrix A is similar to a diagonal matrix
if all eigenvalues are greater than 1, the origin is a
repeller
if all eigenvalues are less than 1, the origin is a
attractor
if some eigenvalues >1 and others <1, the origin is a
saddle point
for complex eigenvalues, if absolute value of eigenvalue > 1
spiral away from origin
for complex eigenvalues, if absolute value of eigenvalue < 1
spiral toward origin
if Re (lambda) > 0 ** differential equations
trajectories spiral outward
if Re (lambda) = 0
trajectories form ellipses
unit vector
vector whose length is 1
two vectors are orthogonal if
their dot product is 0
orthogonal complement
set of all vectors z that are orthogonal to every vector in subspace W
orthonormal set
orthogonal set of unit vectors
symmetric matrix
matrix such that the transpose of A = A
orthogonal matrix
inverse of P = transpose of P; orthonormal columns
spectrum
set of eigenvalues of a matrix
positive definite
Q(x) > 0 for all x =/= 0
negative definite
Q(x) < 0 for all x =/= 0
indefinite
Q(x) <0 and >0 for all x =/= 0
positive semidefinite
Q(x) >= 0 for all x
negative semidefinite
Q(x) <= 0 for all x
singular values of A
square roots of the eigenvalues of A
