Algebra I

Question 1

If v and v’ are column vectors using the bases e₁, e₂,,,,,e_n and e’₁, e’₂,,,,,,e’_n what is the relationship

Answer 1

Pv’ = v where P is an invertible n x n matrix

Question 2

Define eigenvector and eigenvalue

Answer 2

Question 3

What is the dimension of the eigenspace, the nullity of T - lamda(I) equal to

Answer 3

the number of linearly independent eigenvectors corresponding to lamda

Question 4

Theorem: Let T: V to V be a linear map. Then the matrix of T is diagonal with respect to some basis of V if and only if V has a basis consisting of eigenvectors of T

Answer 4

Question 5

Theorem: Let lamda₁,………, lamda_r be distinct eigenvalues of T: V to V ,and v₁,……….,v_r corresponding eigenvectors. Then they’re linearly independent

Answer 5

Question 6

State and prove the Cayley-Hamilton theorem

Answer 6

Question 7

What is denoted by K[x]

Answer 7

The set of polynomials in a single variable x with coefficients in K

Question 8

Define monic

Answer 8

A polynomial with coefficients in a field K is called monic if the coefficient of the highest power of x is 1

Question 9

Theorem: Let A be an n x n matrix over K representing the linear map T:V to V. Then

i) there is a unique monic non-zero polynomical p(x) with minimal degree and coefficients in K such that p(A)=0
ii) if q(x) is any polynomial with q(A)=0, p divides q

Answer 9

Question 10

Define the minimal polynomial

Answer 10

Question 11

How do you calculate the minimal polynomial

Answer 11

Calculate the minimal polynomial for all vectors in the basis, by calculating v, T(v) T²(v),… and stopping when it becomes linearly independent. Then the minimal polynomial of A is the lcm of all the vectors.

Question 12

Define a Jordan chain

Answer 12

Question 13

Define the generalised eigenspace of index i with respect to lamda

Answer 13

Question 14

Define a Jordan block

Answer 14

Question 15

If P is the matrix having the Jordan basis as columns, what is P^-1 AP

Answer 15

J

Question 16

Define Jordan Basis

Answer 16

A Jordan basis is a basis of C^n,1 which is a disjoint union of jordan chians

Question 17

How do we calculate the JCF when n=2 and we have 2 distinct eigenvalues

Answer 17

JCF is J_{lamda1, 1} + J_{lamda2, 1}

C_A(x) = (lamda1 - x)(lamda 2 - x) = mu_A(x)

Question 18

How do we calculate the JCF when n=2 and we have a single eigenvalue lamda

Answer 18

Question 19

How do we calculate the JCF when n=3 and we have 3 distinct eigenvalues

Answer 19

JFC is J_{lamda1, 1} + J_{lamda2, 1} + J_{lamda3, 1}

C_A(x) = (lamda1 - x)(lamda2 - x)(lamda3 - x) = mu_A(x)

Question 20

How do we calculate the JCF when n=3 and we have 2 eigenvalues

Answer 20

Question 21

How do we calculate the JCF when n=3 and we have 1 eigenvalue

Answer 21

Question 22

Define a bilinear map on V and W

Answer 22

Question 23

What is a bilinear form on V

Answer 23

a map t: V x V to K

Question 24

When are symmetric matrices A and B congruent

Answer 24

if there exists an invertible matrix P with B = P^TAP

Question 25

When is a bilinear form on V symmetric

Answer 25

if t(w,v) = t(v,w) for all v,w in V

Question 26

Define a quadratic form

Answer 26

Question 27

Answer 27

Question 28

State Sylvesters Theorem

Answer 28

Question 29

Answer 29

Question 30

When is a quadratic form positive definite

Answer 30

When q(v) \> 0 for all v in V not zero

Question 31

When is V over R a Euclidean space

Answer 31

When t is a positive definite symmetric bilinear form

Question 32

When is a linear map T: V x V orthogonal

Answer 32

if it preserves the scalar product ie T(v) . T(w) = v . w for all v,w in V

Question 33

When is a matrix A orthogonal

Answer 33

When A^TA = I_n

Question 34

# Define an orthonormal basis

Answer 34

Question 35

State the Gram-Schmidt Theorem

Answer 35

Question 36

Answer 36

Question 37

Answer 37

Question 38

Describe the curves for n=2

Answer 38

Question 39

Describe these curves for n=3

Answer 39

Question 40

What is meant by A^star

Answer 40

A conjugate transpose

Question 41

Define the standard inner product on Cⁿ

Answer 41

v.w = v^starw

Question 42

When is a linear map T:Cⁿ to Cⁿ unitary

Answer 42

when it preserves the standard inner product, ie T(v).T(w) = v^starw for all v,w in V

Question 43

When is a matrix A i) unitary ii) Hermitian iii) normal

Answer 43

i) A^starA = I_n ii) A = A^star iii) AA^star = A^starA

Question 44

Write down the symmetric matrix that represents 3x² + 7xy + y²

Answer 44

Question 45

Answer 45

Question 46

Find orthogonal matrix P such that P^TAP is diagonal for A = (-5 12) ( 12 5)

Answer 46

Question 47

Prove that the eigenvalues of a complex Hermitian matrix are all real

Answer 47

Question 48

Define a cyclic group

Answer 48

Question 49

What is meant by Z_n

Answer 49

The group of integers modulo n

Question 50

Define an isomorphism

Answer 50

Question 51

Proposition: Any cyclic group G is isomorphic either to Z or Z_n

Answer 51

Question 52

Define the order of an element g

Answer 52

Question 53

When is a group G spanned by a subset X of G

Answer 53

Question 54

Define Z₄ + Z₆

Answer 54

Question 55

Define the Coset

Answer 55

Question 56

What 3 statements are equivalent for g,k in G

Answer 56

k in H + g H + g = H + k k - g in H

Question 57

What is the relationship between two cosets

Answer 57

They're either equal or disjoint

Question 58

State Lagranges Theorem

Answer 58

Let G be a finite (abelian) group and H a subgroup of G. Then the order of H divides the order of G

Question 59

Define the index of H in G

Answer 59

The number of distinct cosets of H in G, written [G : H]

Question 60

Proposition: Let G be a finite (abelian) group. Then for any g in G, the order of g divides the order of G

Answer 60

Question 61

Define the sum of subsets A + B

Answer 61

Question 62

If H is a subgroup and H + g, H + h cosets of G, what is (H + h) + (H + g)

Answer 62

H + (h + g)

Question 63

Define the quotient group G/H

Answer 63

The group of cosets H + g of H in G

Question 64

Define a homomorphism

Answer 64

Question 65

Define the kernel for a homomorphism

Answer 65

Question 66

When is a homomorphism i) a monomorphism ii) an epimorphism

Answer 66

i) when its injective ii) when its surjective

Question 67

Define a free abelian group of rank n

Answer 67

Question 68

When do elements x₁,.......,x_n form a free basis of the abelian group G

Answer 68

if and only if they're linearly independent

Question 69

Proposition: An abelian group G is a free abelian if and only if it has a free basis x₁,........,x_n, in which case there is an isomorphism from phi: G to Z_n with phi(x_i) = x_i for i in {1,.....,n}

Answer 69

Question 70

Answer 70

Question 71

When is a n x n matrix in Z unimodular

Answer 71

When det(A) = plus or minus 1

Question 72

What are (UR1), (UR2), (UR3)?

Answer 72

UR1 - replace row r_i by r_i + tr_j for j not equal to i UR2 - interchange two rows UR3 - replace a row r_i by -r_i

Question 73

When is a m x n matrix with rank r in Smith Normal form?

Answer 73

When b_ii = d_i for 1 \<= i \<= r, b_ij = 0 for all i not equal to j, and d_i divides d_i+1

Question 74

What is the top left entry of a matrix in SNF

Answer 74

The highest common factor of all non-zero entries

Question 75

Write down the matrix corresponding to \< x₁ x₂ x₃ | x₁ + 3x₂ - x₃, 2x₁ + x₃\>

Answer 75

Question 76

State the fundamental theorem of finitely generated abelian groups

Answer 76

Question 77

If a matrix A in SNF form is (7 0) (0 21) what is G isomoprhic to

Answer 77

Z₇ + Z₂₁

Question 78

When n = 36, what could G be isomorphic to

Answer 78

Z₃₆ Z₂ x Z₁₈ Z₃ x Z₁₂ Z₆ x Z₆