MML — Linear Algebra Flashcards
Regular/nonsingular matrices
alies for invertible matrices
What is a singular matrix
is an noninvertible matrix
What is a special solution of a linear system
it’s its particular solution (others can exist)
A pivot of a row
first non-zero coefficient
A matrix is in row-echelon form if
1) and 2)
1) All zero-rows are on the bottom of the matrix
2) If the row is not first, its pivot would be to the right of the upper row’s pivot.
A matrix is in reduced row-echelon form if
1) , 2) and 3)
1) It’s in row-echelon form
2) Every pivot is 1
3) The pivot is the only non-zero entry in its column
Four axioms of a Group
1) Closure
2) Associativity
3) Neutral element
4) Inverse element
Abelian group
is a commutative group
General Linear Group
The set of regular (invertible) matrices n⨉n is a group with respect to matrix multiplication and is called general linear group GL(n,ℝ). However, since matrix multiplication is not commutative, the group is not Abelian
Axioms of a vector space using groups
Assuming addition of vectors and scaling with scalars to be correctly defined.
1) The set of vectors is an Abelian group with respect to addition of vectors.
2) Two distributivity axioms
3) Associativity (vector and scalars)
4) Neutral element with respect to scalar multiplication (so-called outer operation while addition is inner operation)
Full rank of a matrix
A matrix A that is m⨉n has full rank if its rank is equal to the largest possible number for a matrix of such dimensions (i.e. rank(A) = min[n, m] )
Linear mapping (or vector space homomorphism/linear transformation)
For vector spaces V and W, a mapping F: V->W is called a linear mapping if
∀x, y ∈ V ∀λ, ψ ∈ ℝ : F(λx + ψy) = λF(x) + ψF(y).
Injective, surjective and bijective mappings
Let F : V->W, V and W are arbitrary sets. Then F is called
1) Injective if ∀x, y ∈ V, F(x) = F(y) => x = y
2) Surjective if F(V) = W
3) Bijective if it’s injective and surjective
Special cases of linear mappings between vector spaces V and W
1) Isomorphism: F:V->W linear and bijective
2) Endomorphism: F:V->V linear
3) Automorphism: F:V->V linear and bijective
Definition of equivalent and similar matrices
Two m×n matrices A and A* are equivalent if there exist regular matrices S — n×n and T — m×m, such that
A* = T^(-1) A S
Two n×n matrices A and A* are called similar if there exists a regular matrix T such that
A* = T^(-1) A T
Similar matrices are always equivalent. However, equivalent matrices are not necessarily similar.
What are domain and codomain of F:V->W?
domain is a kernel (null space) and codomain is the image (range) of F.
If a kernel of a linear mapping F = {0} we can say for sure
that F is injective (one-to-one).
Column space of a matrix of a linear mapping
is its image.
Affine mapping
For two vector spaces V and W, a linear mapping F: V->W, and a∈W, the mapping
f: V->W that is
x->a + F(x)
is an affine mapping from V to W. The vector a is called the translation vector of f.
Hadamard product
Element-wise multiplication of matrices.
A generating set of a vector space
A set of vectors is called a generating set if every vector from the space can be represented as a linear combination of vectors from this set. We write V = span[x_1, …, x_k]
Rank deficient matrix
A matrix is said to be rank deficient if it does not have full rank
