a system of linear equations with at least one solution
New cards
2
inconsistent system
a system of linear equations with no solution
New cards
3
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
New cards
4
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
New cards
5
spanning set
the set of all linear combinations of a set of vectors in Rn
New cards
6
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
New cards
7
symmetric matrix
a matrix whose transpose is equal to itself
New cards
8
elementary matrix
a matrix that is formed by performing one elementary row operation on the identity matrix
New cards
9
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
New cards
10
subspace
a set of vectors such that zero belongs to the set and the set is closed under linear combinations
New cards
11
row space
the subspace of Rn spanned by the rows of A
New cards
12
column space
the subspace of Rn spanned by the columns of A
New cards
13
null space
the subspace spanned by solutions to the equation A x = 0
New cards
14
basis
a set of vectors in S such that the vectors span S and are linearly independent
New cards
15
standard basis
the standard unit vectors for Rn
New cards
16
the basis theorem
any two bases for S have the same number of vectors
New cards
17
dimension
the number of vectors in a basis of a subspace
New cards
18
rank
the dimension of the row and column spaces of a matrix
New cards
19
nullity
the dimension of a null space
New cards
20
rank nullity theorem
the rank and nullity of a matrix sum to five the number of columns
New cards
21
kernel
the set of vectors that are sent to the zero vector by a linear transformation
New cards
22
range
the set of vectors that are images of vectors in the domain of a linear transformation
New cards
23
eigenvalue
a scalar that multiplies a vector and gives the same result as a matrix A multiplying the vector
New cards
24
eigenvector
vector corresponding to an eigenvalue
New cards
25
algebraic multiplicity
power of the eigenvalue in the characteristic polynomial
Studied by 12 people
