Mathematical methods UPDATED ACTUAL
QUESTIONS AND CORRECT ANSWERS
Dual vector space V* of V - Correct answers✔✔Set of all linear transformations that map from
vector space V to scalars (Real numbers)
Criteria of vector sub space - Correct answers✔✔1. Closed under scalar multiplication
2. Closed under vector addition
Define convergence (formally and intuitively) - Correct answers✔✔1. In normed vector space C,
sequence converges to vector v if for every epsilon>0, there exists a positive integer k such that
v_i is in a ball centred on v with radius epsilon for all i>k
2. The tail of the sequence is sufficiently small to fit into small gaps around v
Define Cauchy convergence (formally and intuitively) - Correct answers✔✔1. For every small
epsilon >0, there is a positive integer k st vi-vj is in the ball centred at 0 with radius epsilon
2. The tail of the sequence is sufficiently small so that the norm between two elements of the
sequence is smaller than epsilon
Intuitive definition of open set - Correct answers✔✔set not including the boundary
Intuitive definition of closed set - Correct answers✔✔if its complement is open
The set contains its boundary
Intuitive definition of limit points - Correct answers✔✔a set's boundaries
,Definition of compact set - Correct answers✔✔set is closed and bounded
Definition of dense subset of V - Correct answers✔✔S is dense in V if every value in V can be
approximated arbitrarily well to an element in S
e.g. Q is dense in R
Definition of separable set - Correct answers✔✔V is separable if it has a subset that is dense and
countable
E.g. R is separable in Q because Q is countable and dense in R
Complete set - Correct answers✔✔Set where every Cauchy sequence converges
Hilbert space - Correct answers✔✔Complete normed vector space with inner product associated
with that norm
Banach space - Correct answers✔✔Complete normed vector space
An orthonormal system in Hilbert space is an orthonormal basis if any of the four statements is
true - Correct answers✔✔1. Every vector in Hilbert space can be written as v = sum(<e_i, v>
e_i)
2. For every vector, a norm is defined as sum(|<e_i, v>|^2)
3. If <e_i, v>=0 for every I then v = 0
4. Closure of the span of orthonormal system = Entire Hilbert space
, Define a compact linear operator T on Hilbert space - Correct answers✔✔for every bounded
sequence v_k in H, the sequence (Tv_k) contains a convergent subsequence
Properties of compact operators and eigenvalues/eigenvectors - Correct answers✔✔1.
Eigenvalues of T are real
2. Eigenvectors of different eigenvalues are orthogonal
3. Set of eigenvalues is finite or an infinite series tending to 0
4. Each non-zero eigenvalue has finite degeneracy
5. Orthonormal system of eigenvectors with non-zero eigenvalues forms basis on Im(T)
Solve (T-lambda I )v = 0 - Correct answers✔✔Solve Ker(T-lambda I) = 0 : determinant =0
If lambda is not one of the eigenvalues, the equation only has trivial solutions
Solve (T-lambda I )v = u - Correct answers✔✔1. Substitute Tv_k = lambda_kv_k
2. (lambda_k - lambda) <e_k, v> = <e_k, u>
3. <e_k, v> = <e_k, u>/(lambda_k - lambda)
If lambda equals one of the eigenvalues lambda_k, solution is sum(alpha_k e_k)
Basis of cosine Fourier series - Correct answers✔✔c_0 = 1/sqrt(a)
c_k = sqrt(2/a)cos(k*pi*x/a)
k = 1, 2, 3,
QUESTIONS AND CORRECT ANSWERS
Dual vector space V* of V - Correct answers✔✔Set of all linear transformations that map from
vector space V to scalars (Real numbers)
Criteria of vector sub space - Correct answers✔✔1. Closed under scalar multiplication
2. Closed under vector addition
Define convergence (formally and intuitively) - Correct answers✔✔1. In normed vector space C,
sequence converges to vector v if for every epsilon>0, there exists a positive integer k such that
v_i is in a ball centred on v with radius epsilon for all i>k
2. The tail of the sequence is sufficiently small to fit into small gaps around v
Define Cauchy convergence (formally and intuitively) - Correct answers✔✔1. For every small
epsilon >0, there is a positive integer k st vi-vj is in the ball centred at 0 with radius epsilon
2. The tail of the sequence is sufficiently small so that the norm between two elements of the
sequence is smaller than epsilon
Intuitive definition of open set - Correct answers✔✔set not including the boundary
Intuitive definition of closed set - Correct answers✔✔if its complement is open
The set contains its boundary
Intuitive definition of limit points - Correct answers✔✔a set's boundaries
,Definition of compact set - Correct answers✔✔set is closed and bounded
Definition of dense subset of V - Correct answers✔✔S is dense in V if every value in V can be
approximated arbitrarily well to an element in S
e.g. Q is dense in R
Definition of separable set - Correct answers✔✔V is separable if it has a subset that is dense and
countable
E.g. R is separable in Q because Q is countable and dense in R
Complete set - Correct answers✔✔Set where every Cauchy sequence converges
Hilbert space - Correct answers✔✔Complete normed vector space with inner product associated
with that norm
Banach space - Correct answers✔✔Complete normed vector space
An orthonormal system in Hilbert space is an orthonormal basis if any of the four statements is
true - Correct answers✔✔1. Every vector in Hilbert space can be written as v = sum(<e_i, v>
e_i)
2. For every vector, a norm is defined as sum(|<e_i, v>|^2)
3. If <e_i, v>=0 for every I then v = 0
4. Closure of the span of orthonormal system = Entire Hilbert space
, Define a compact linear operator T on Hilbert space - Correct answers✔✔for every bounded
sequence v_k in H, the sequence (Tv_k) contains a convergent subsequence
Properties of compact operators and eigenvalues/eigenvectors - Correct answers✔✔1.
Eigenvalues of T are real
2. Eigenvectors of different eigenvalues are orthogonal
3. Set of eigenvalues is finite or an infinite series tending to 0
4. Each non-zero eigenvalue has finite degeneracy
5. Orthonormal system of eigenvectors with non-zero eigenvalues forms basis on Im(T)
Solve (T-lambda I )v = 0 - Correct answers✔✔Solve Ker(T-lambda I) = 0 : determinant =0
If lambda is not one of the eigenvalues, the equation only has trivial solutions
Solve (T-lambda I )v = u - Correct answers✔✔1. Substitute Tv_k = lambda_kv_k
2. (lambda_k - lambda) <e_k, v> = <e_k, u>
3. <e_k, v> = <e_k, u>/(lambda_k - lambda)
If lambda equals one of the eigenvalues lambda_k, solution is sum(alpha_k e_k)
Basis of cosine Fourier series - Correct answers✔✔c_0 = 1/sqrt(a)
c_k = sqrt(2/a)cos(k*pi*x/a)
k = 1, 2, 3,
