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
Define an orthonormal basis
State the Gram-Schmidt Theorem
Describe the curves for n=2
Describe these curves for n=3
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
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
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
Define a homomorphism
Define the kernel for a homomorphism
Define a free abelian group of rank n
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}
Write down the matrix corresponding to < x1 x2 x3 | x1 + 3x2 - x3, 2x1 + x3>
State the fundamental theorem of finitely generated abelian groups
