Linear Algebra //

Question 1

Field

Answer 1

A set S on which addition and multiplication are defined is called a field if it satisfies each of the axioms A1-4, M1-4 and D, and if, in addition 1 ≠ 0

Question 2

Vector space

Answer 2

A vector space over a field K is a set V which has two basic operations, addition and scalar multiplication. Thus for every pair u,v ∈ V, u + v ∈ V is defined, and for every α ∈ K, αv ∈ V is defined. The following axioms are satisfied for all α,β ∈ K and all u,v ∈ V

i) vector addition satisfies axioms A1-4
ii) α(u + v) = αu + αv
iii) (α + β)v = αv + βv
iv) (αβ)v = α(βv)
v) 1v = v

Question 3

Scalar

Answer 3

An element of the field K

Question 4

Linearly dependent

Answer 4

Let V be a vector space over the field K. The vectors v1, v2, … vn are said to be linearly dependent if there exist scalars α1, α2, …, αn ∈ K, not all 0, such that α1v1 + α2v2 + … + αnvn = 0

Question 5

Linearly independent

Answer 5

v1, v2, … vn are linearly independent if the only scalars α1, α2, …, αn ∈ K that satisfy α1v1 + α2v2 + … + αnvn = 0 are α1 = α2 = … = αn = 0

Question 6

Linear combination

Answer 6

Vectors of the form α1v1 + α2v2 + … + αnvn for α1, α2, …, αn ∈ K are called linear combinations of v1, v2, …, vn

Question 7

Span

Answer 7

The vectors v1, v2, …, vn in V span V if every vector v ∈ V is a linear combination α1v1 + α2v2 + … + αnvn of v1, …, vn

Question 8

Basis

Answer 8

The vectors v1, …, vn in V form a basis of V if they are linearly independent and span V

Question 9

Subspace

Answer 9

A subspace of V is a non-empty subset W ⊂ V such that where u, v ∈ W and α ∈ K, u + v ∈ W and αv ∈ W

Question 10

The set W1 + W2

Answer 10

Let W1, W2 be subspaces of the vector space V. Then W1 + W2 = {w1 + w2 | w1 ∈ W1, w2 ∈ W2}

Question 11

Linear map

Answer 11

Let U, V be two vector spaces over the same field K. A linear map T from U to V is a function T: U → V such that

i) T(u1 + u2) = T(u1) + T(u2) for all u1,u2 ∈ U;
ii) T(αu) = αT(u) for all α ∈ K and u ∈ U

Question 12

Image

Answer 12

Let T: U → V be a linear map. The image of T, written as im(T) is defined to be the set of vectors v ∈ V such that v = T(u) for some u ∈ U

Question 13

Kernel

Answer 13

Let T: U → V be a linear map. The kern of T, written as ker(T), is defined to be the set of vectors u ∈ U such that T(u) = 0v

Question 14

Rank

Answer 14

dim(im(T))

Question 15

Nullity

Answer 15

dim(ker(T))

Question 16

Non-singular linear map

Answer 16

A linear map T such that dim(U) = dim(V) = n, such that T is bijective, rank(T) = n and nul(T) = 0

Question 17

Addition on linear maps T1 + T2: U → V

Answer 17

(T1 + T2)(u) = T1(u) + T2(u) for u ∈ U

Question 18

Scalar multiplication αT1: U → V

Answer 18

(αT1)(u) = αT1(u) for u ∈ U

Question 19

Composition T2T1: U → W, where T1: U → V and T2: V → W

Answer 19

(T2T2)(u) = T2(T1(u)) for u ∈ U

Question 20

Hom_K(U, V)

Answer 20

Hom_K(U,V) = {T: U → V | T is linear}

Question 21

Multiplication of matrices AB

Answer 21

Let A = (αij) be an lxm matrix over K and let B = (βij) be an mxn matrix over K.
The product AB = C = (γij) is an lxn matrix where for 1 ≤ i ≤ l and 1 ≤ j ≤ n,
γij = Σ(k=1,m) αik*βkj

note: #cols of A = #rows of B

Question 22

Nullspace of a matrix A

Answer 22

Given a matrix A, the set of column vectors x ∈ K^n,1 solving Ax = 0 is the nullspace of A

Question 23

Elementary row operations R1, R2, R3

Answer 23

R1 for some i ≠ j, add a multiple of rj to ri
R2 interchange two rows
R3 multiply a row by a non-zero scalar

Question 24

Upper echelon form

Answer 24

i) all zero rows are below all non zero rows
ii) let r1,…,rs be the non-zero rows. Then each ri with 1 ≤ i ≤ s has 1 as its first non-zero entry
iii) c(1) < c(2) < … < c(s)
iv) α_k,c(i) = 0 for all k > i

Question 25

Row reduced form

Answer 25

i) all zero rows are below all non zero rows
ii) let r1,…,rs be the non-zero rows. Then each ri with 1 ≤ i ≤ s has 1 as its first non-zero entry
iii) c(1) < c(2) < … < c(s)
iv) α_k,c(i) = 0 for all k ≠ i

Question 26

Smith normal form

Answer 26

A matrix in the form
(I_s | 0_s,n-s
0_m-s,s | 0_m-s,n-s)
I_s denotes the sxs identity matrix, and 0_k,l denotes the kxl zero matrix

Question 27

Row-space of A

Answer 27

The subspace of K^n spanned by the rows r1,…,rm of A

Question 28

Row rank of A

Answer 28

The dimension of the row-space of A/the size of the largest LI subset of r1,…,rm

Question 29

Column-space

Answer 29

The subspace of K^m,1 spanned by the columns c1,…,cn of A

Question 30

Column rank

Answer 30

The dimension of the column-space of A/the size of the largest LI subset of c1,…,cn

Question 31

Invertible linear map T

Answer 31

T is invertible if there is a map T^-1: V → U with TT^-1 = Iv and T^-1T = Iu

Question 32

Invertible matrix A

Answer 32

A is invertible if AA^-1 = Im and A^-1A = In

Question 33

Elementary row matrices corresponding to R1, R2, R3

Answer 33

1. E(n)(λ,i,j, 1) (where i ≠ j) is the nxn matrix equal to the identity, but with an additional non-zero entry λ in the (i,j) position
2. E(n)(i,j, 2) is the nxn matrix with its ith and jth rows interchanged
3. E(n)(λ,i, 3) (where λ ≠ 0) is the nxn identity matrix with its (i,i) entry replaced by λ

Question 34

Sigma version of det(A)

Answer 34

Σ(φ ∈ Sn) sign(φ)α_1φ(1)α_2φ(2)…α_nφ(n)

Question 35

Upper triangular matrix

Answer 35

All of its entries below the main diagonal are zero

Question 36

Diagonal matrix

Answer 36

All entries not on the main diagonal are zero

Question 37

Transpose A^T of A

Answer 37

Let A = (αij) be an mxn matrix, the transpose A^T of A js the nxm matrix (βij), where βij = αji for 1 ≤ i ≤ n, 1 ≤ j ≤ m

Question 38

Minor Mij of the nxn matrix A

Answer 38

The minor Mij is the determinant of the (n-1)x(n-1) matrix obtained from A by deleting the ith row and jth column of A

Question 39

(i,j)-th co-factor cij of A

Answer 39

cij = (-1)^(i+j)*Mij

Question 40

Adjoint matrix adj(A) of the nxn matrix A

Answer 40

adj(A) is the nxn matrix where the (i,j)-th element is the cofactor cji (ie the transpose of the matrix of cofactors)

Question 41

Equivalent matrices

Answer 41

Two matrices A and B are said to be equivalent if there exist invertible P and Q with B = QAP (represent the same linear map)

Question 42

Similar matrices

Answer 42

Two nxn matrices over K are said to be similar if there exists and nxn invertible matrix P with B = P^-1AP (represent the same linear map from V to V wrt different bases of V)

Question 43

Eigenvector/value of T

Answer 43

Suppose that for some non-zero vector v ∈ V and some scalar λ ∈ K, we have T(v) = λv. Then v is called an eigenvector of T, and λ is called the eigenvalue of T corresponding to v

Question 44

Characteristic equation of a matrix A

Answer 44

For an nxn matrix A, the characteristic equation is det(A - xIn) = 0

Question 45

Scalar product of two vectors

Answer 45

If v = (α1, …, αn), w = (β1, …, βn)

v.w = Σ(i=1, n)αiβi

Question 46

Orthonormal basis

Answer 46

A basis b1, …, bn of ℝ^n is orthonormal if

i) bi.bi = 1 for 1 ≤ i ≤ n
ii) bi.bj = 0 for 1 ≤ i,j ≤ n and i ≠ j

Question 47

Symmetric matrix A

Answer 47

An nxn matrix A such that A^T = A

Question 48

Orthogonal

Answer 48

An nxn matrix A is orthogonal if A^T = A^-1

Question 49

Diagonalisable

Answer 49

A matrix which is similar to a diagonal matrix

Question 50

Complementary subspaces

Answer 50

Two subspaces W1 and W2 of V are complementary if W1 ∩ W2 = {0} and W1 + W2 = V