MML — Linear Algebra

Question 1

Regular/nonsingular matrices

Answer 1

alies for invertible matrices

Question 2

What is a singular matrix

Answer 2

is an noninvertible matrix

Question 3

What is a special solution of a linear system

Answer 3

it’s its particular solution (others can exist)

Question 4

A pivot of a row

Answer 4

first non-zero coefficient

Question 5

A matrix is in row-echelon form if

Answer 5

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.

Question 6

A matrix is in reduced row-echelon form if

Answer 6

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

Question 7

Four axioms of a Group

Answer 7

1) Closure
2) Associativity
3) Neutral element
4) Inverse element

Question 8

Abelian group

Answer 8

is a commutative group

Question 9

General Linear Group

Answer 9

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

Question 10

Axioms of a vector space using groups

Answer 10

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)

Question 11

Full rank of a matrix

Answer 11

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] )

Question 12

Linear mapping (or vector space homomorphism/linear transformation)

Answer 12

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).

Question 13

Injective, surjective and bijective mappings

Answer 13

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

Question 14

Special cases of linear mappings between vector spaces V and W

Answer 14

1) Isomorphism: F:V->W linear and bijective
2) Endomorphism: F:V->V linear
3) Automorphism: F:V->V linear and bijective

Question 15

Definition of equivalent and similar matrices

Answer 15

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.

Question 16

What are domain and codomain of F:V->W?

Answer 16

domain is a kernel (null space) and codomain is the image (range) of F.

Question 17

If a kernel of a linear mapping F = {0} we can say for sure

Answer 17

that F is injective (one-to-one).

Question 18

Column space of a matrix of a linear mapping

Answer 18

is its image.

Question 19

Affine mapping

Answer 19

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.

Question 20

Hadamard product

Answer 20

Element-wise multiplication of matrices.

Question 21

A generating set of a vector space

Answer 21

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]

Question 22

Rank deficient matrix

Answer 22

A matrix is said to be rank deficient if it does not have full rank