Linear Algebra 2B

consistent system

a system of linear equations with at least one solution

inconsistent system

a system of linear equations with no solution

row echelon form

a matrix such that:
any all zero rows are on the bottom
all leading entries are to the left of any leading entries below them

reduced row echelon form

a matrix such that:
any all zero rows are on the bottom
all leading entries are to the left of any leading entries below them
all leading entries are 1s
any column with a leading 1 has zeroes elsewhere

spanning set

the set of all linear combinations of a set of vectors in Rn

linearly independent

a set of vectors such that the only solution to the linear combination of all vectors equal to the zero vector is the trivial solution

symmetric matrix

a matrix whose transpose is equal to itself

elementary matrix

a matrix that is formed by performing one elementary row operation on the identity matrix

fundamental theorem of invertible matrices

if A is a n x n matrix, the following are equivalent
A is invertible
A x = b has a unique solution for all b in Rn
A x = 0 has only the trivial solution
the reduced echelon form of A is the identity
A is a product of elementary matrices

subspace

a set of vectors such that zero belongs to the set and the set is closed under linear combinations

row space

the subspace of Rn spanned by the rows of A

column space

the subspace of Rn spanned by the columns of A

null space

the subspace spanned by solutions to the equation A x = 0

basis

a set of vectors in S such that the vectors span S and are linearly independent

standard basis

the standard unit vectors for Rn

the basis theorem

any two bases for S have the same number of vectors

dimension

the number of vectors in a basis of a subspace

rank

the dimension of the row and column spaces of a matrix

nullity

the dimension of a null space

rank nullity theorem

the rank and nullity of a matrix sum to five the number of columns

kernel

the set of vectors that are sent to the zero vector by a linear transformation

range

the set of vectors that are images of vectors in the domain of a linear transformation

eigenvalue

a scalar that multiplies a vector and gives the same result as a matrix A multiplying the vector

eigenvector

vector corresponding to an eigenvalue

algebraic multiplicity

power of the eigenvalue in the characteristic polynomial

geometric multiplicity

dimension of eigenspace

similar matrix

a matrix such that A = P-1 B P