Algebra I Flashcards
If v and v’ are column vectors using the bases e1, e2,,,,,en and e’1, e’2,,,,,,e’n what is the relationship
Pv’ = v where P is an invertible n x n matrix
Define eigenvector and eigenvalue
What is the dimension of the eigenspace, the nullity of T - lamda(I) equal to
the number of linearly independent eigenvectors corresponding to lamda
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
Theorem: Let lamda1,………, lamdar be distinct eigenvalues of T: V to V ,and v1,……….,vr corresponding eigenvectors. Then they’re linearly independent
State and prove the Cayley-Hamilton theorem
What is denoted by K[x]
The set of polynomials in a single variable x with coefficients in K
Define monic
A polynomial with coefficients in a field K is called monic if the coefficient of the highest power of x is 1
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
Define the minimal polynomial
How do you calculate the minimal polynomial
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.
Define a Jordan chain
Define the generalised eigenspace of index i with respect to lamda
Define a Jordan block
If P is the matrix having the Jordan basis as columns, what is P-1 AP
J
Define Jordan Basis
A Jordan basis is a basis of Cn,1 which is a disjoint union of jordan chians
How do we calculate the JCF when n=2 and we have 2 distinct eigenvalues
JCF is Jlamda1, 1 + Jlamda2, 1
CA(x) = (lamda1 - x)(lamda 2 - x) = muA(x)
How do we calculate the JCF when n=2 and we have a single eigenvalue lamda
How do we calculate the JCF when n=3 and we have 3 distinct eigenvalues
JFC is Jlamda1, 1 + Jlamda2, 1 + Jlamda3, 1
CA(x) = (lamda1 - x)(lamda2 - x)(lamda3 - x) = muA(x)
How do we calculate the JCF when n=3 and we have 2 eigenvalues
How do we calculate the JCF when n=3 and we have 1 eigenvalue
Define a bilinear map on V and W
What is a bilinear form on V
a map t: V x V to K
When are symmetric matrices A and B congruent
if there exists an invertible matrix P with B = PTAP
When is a bilinear form on V symmetric
if t(w,v) = t(v,w) for all v,w in V
Define a quadratic form
State Sylvesters Theorem
When is a quadratic form positive definite
When q(v) > 0 for all v in V not zero
When is V over R a Euclidean space
When t is a positive definite symmetric bilinear form
When is a linear map T: V x V orthogonal
if it preserves the scalar product ie T(v) . T(w) = v . w for all v,w in V
When is a matrix A orthogonal
When ATA = In
Define an orthonormal basis
State the Gram-Schmidt Theorem
Describe the curves for n=2
Describe these curves for n=3
What is meant by Astar
A conjugate transpose
Define the standard inner product on Cn
v.w = vstarw
When is a linear map T:Cn to Cn unitary
when it preserves the standard inner product, ie T(v).T(w) = vstarw for all v,w in V
When is a matrix A
i) unitary
ii) Hermitian
iii) normal
i) AstarA = In
ii) A = Astar
iii) AAstar = AstarA
Write down the symmetric matrix that represents 3x2 + 7xy + y2
Find orthogonal matrix P such that PTAP is diagonal for A = (-5 12)
( 12 5)
Prove that the eigenvalues of a complex Hermitian matrix are all real
Define a cyclic group
What is meant by Zn
The group of integers modulo n
Define an isomorphism
Proposition: Any cyclic group G is isomorphic either to Z or Zn
Define the order of an element g
When is a group G spanned by a subset X of G
Define Z4 + Z6
Define the Coset
What 3 statements are equivalent for g,k in G
k in H + g
H + g = H + k
k - g in H
What is the relationship between two cosets
They’re either equal or disjoint
State Lagranges Theorem
Let G be a finite (abelian) group and H a subgroup of G. Then the order of H divides the order of G
Define the index of H in G
The number of distinct cosets of H in G, written
[G : H]
Proposition: Let G be a finite (abelian) group. Then for any g in G, the order of g divides the order of G
Define the sum of subsets A + B
If H is a subgroup and H + g, H + h cosets of G, what is (H + h) + (H + g)
H + (h + g)
Define the quotient group G/H
The group of cosets H + g of H in G
Define a homomorphism
Define the kernel for a homomorphism
When is a homomorphism
i) a monomorphism
ii) an epimorphism
i) when its injective
ii) when its surjective
Define a free abelian group of rank n
When do elements x1,…….,xn form a free basis of the abelian group G
if and only if they’re linearly independent
Proposition: An abelian group G is a free abelian if and only if it has a free basis x1,……..,xn, in which case there is an isomorphism from phi: G to Zn with phi(xi) = xi for i in {1,…..,n}
When is a n x n matrix in Z unimodular
When det(A) = plus or minus 1
What are (UR1), (UR2), (UR3)?
UR1 - replace row ri by ri + trj for j not equal to i
UR2 - interchange two rows
UR3 - replace a row ri by -ri
When is a m x n matrix with rank r in Smith Normal form?
When bii = di for 1 <= i <= r, bij = 0 for all i not equal to j, and di divides di+1
What is the top left entry of a matrix in SNF
The highest common factor of all non-zero entries
Write down the matrix corresponding to < x1 x2 x3 | x1 + 3x2 - x3, 2x1 + x3>
State the fundamental theorem of finitely generated abelian groups
If a matrix A in SNF form is
(7 0)
(0 21)
what is G isomoprhic to
Z7 + Z21
When n = 36, what could G be isomorphic to
Z36
Z2 x Z18
Z3 x Z12
Z6 x Z6
