Assume a space-time n the dimension and g the metric in said space-time. (a) Show that it is alw...
Assume a space-time n the dimension and g the metric in said space-time.
(a) Show that it is always possible to find an orthonormal basis {e1, e2, ... en}, such that:
hint, use induction.
b) Show that the signature of the metric is
independent of the chosen orthonormal basis.
Homework Answers
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Assume a space-time n the dimension and g the metric in said space-time. (a) Show that it is alw...
Let m, n EN\{1}, V be a vector space over R of dimension n and (v1,...
Let m, n EN\{1}, V be a vector space over R of dimension n and (v1, ..., Vm) be an m tuple of V. (Select ALL that are TRUE) If m > n then (v1, ..., Vm) spans V. If (v1, ..., Um) is linearly independent then m <n. (v1, ..., Um) is linearly dependent if and only if for all i = 1,..., m we have that U; Espan(vi, .., Vi-1, Vj+1, ..., Um). Assume there exists exactly one...
(5) Here is a fascinating equivalence for being a complete metric space that we will use...
(5) Here is a fascinating equivalence for being a complete metric space that we will use later. Let (X,d) be a metric space. (b) ** (10 points) Show that the following are equivalent: • (X, d) is complete; • for every family of non-empty closed subsets Fo, F1, F2, ... of X such that F, 2 F12 F22... and limn700 diam( Fn) = 0, it holds that Nnen Fn = {a} for some a € X. (Hint: for the reverse...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Def...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...
2. Let ro < 1<..< n be n + 1 distinct points in IR. Define polynomials...
2. Let ro < 1<..< n be n + 1 distinct points in IR. Define polynomials Co, ..., (n of degree n by (r - k) Let P, = 1,[r] be the polynomials of degree n, which is a vector space of dimension n + 1. (a) Show that the n+1 polynomials {lo, ..., Ln^ are basis for P i.e., they are linearly independent. (b) Find the coordinates [f]в of polynomial f E 1, with respect to the basis l-[10,...
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ]...
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ] -» R, equipped with the inner product f(t)g(t)d (f,g) = (a) Let UC V be the subspace spanned by B = (sinr, cos x, 1) (you may assume without proof that B is linearly independent, and hence a basis for U). Find the B-matrix [D]93 of the "derivative linear transformation" D : U -> U given by D(f) = f'. (b) Let WC V...
Problem 4 Let hn] be the sequence whose Fourier transform H(w) is real and as follows and let g[n] = (-1)"h[n] a-3 pts) Plot G(w) for w E-π, π]. Detail your derivations. Make sure to show the max...
Problem 4 Let hn] be the sequence whose Fourier transform H(w) is real and as follows and let g[n] = (-1)"h[n] a-3 pts) Plot G(w) for w E-π, π]. Detail your derivations. Make sure to show the maximuin value of G(w) b - [2 pts| Derive explicitly the impulse response of the following system n] Hint: Besides some graphical consideration, there is no calculation. The answer is mostly based orn the use of properties. c - 3 pts] Up to...
New problems for 2020 1. A topological space is called a T3.space if it is a...
New problems for 2020 1. A topological space is called a T3.space if it is a T, space and for every pair («,F), where € X and F(carefull), there is a continuous function 9 :X (0,1 such that f(x) 0 and f =1 on F. Prove that such a space has the Hausdorff Separation Property. (Hint: One point subsets are closed.] 2. Let X be topological space, and assume that both V and W are subbases for the topology. Show...
Sur I Nano 2019/20) Qunntum physics exercices 1.1 Linear algebra and formalism Exercice 1.1.1 Basic calculations...
Sur I Nano 2019/20) Qunntum physics exercices 1.1 Linear algebra and formalism Exercice 1.1.1 Basic calculations We consideran llibert prace Ey of dimension 2 and the two following vectors of En = (17.) and le >= () acting on vectors of Eh 21- We consider also the linear operator = 1+5 1 Calculate the square norms of the two vectors < Hermitian scalar products <ul> and < > 2. Calculate the eigensalues and cigarvectors of A. >, < > and...
Please only solve part C Assume the following state space representation of a discrete-time servomotor system. (As a review for the Final Exam, you might check this state space representation with th...
Please only solve part C
Assume the following state space representation of a discrete-time servomotor system. (As a review for the Final Exam, you might check this state space representation with the difference equation in Problem 1 on Homework 2. This parenthetical comment is not a required part for Homework 8.) 2. 0.048371 u(n) 1.9048x(n) lo.04679 [1,0]x(n) y(n) Compute the open-loop eigenvalues of the system. That is, find the eigenvalues of Ф. Check controllability of the system. Or, answer the...
Let m, n EN\{1}, V be a vector space over R of dimension n and (v1, ..., Vm) be an m tuple of V. (Select ALL that are TRUE) If m > n then (v1, ..., Vm) spans V. If (v1, ..., Um) is linearly independent then m <n. (v1, ..., Um) is linearly dependent if and only if for all i = 1,..., m we have that U; Espan(vi, .., Vi-1, Vj+1, ..., Um). Assume there exists exactly one...
(5) Here is a fascinating equivalence for being a complete metric space that we will use later. Let (X,d) be a metric space. (b) ** (10 points) Show that the following are equivalent: • (X, d) is complete; • for every family of non-empty closed subsets Fo, F1, F2, ... of X such that F, 2 F12 F22... and limn700 diam( Fn) = 0, it holds that Nnen Fn = {a} for some a € X. (Hint: for the reverse...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...
2. Let ro < 1<..< n be n + 1 distinct points in IR. Define polynomials Co, ..., (n of degree n by (r - k) Let P, = 1,[r] be the polynomials of degree n, which is a vector space of dimension n + 1. (a) Show that the n+1 polynomials {lo, ..., Ln^ are basis for P i.e., they are linearly independent. (b) Find the coordinates [f]в of polynomial f E 1, with respect to the basis l-[10,...
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ] -» R, equipped with the inner product f(t)g(t)d (f,g) = (a) Let UC V be the subspace spanned by B = (sinr, cos x, 1) (you may assume without proof that B is linearly independent, and hence a basis for U). Find the B-matrix [D]93 of the "derivative linear transformation" D : U -> U given by D(f) = f'. (b) Let WC V...
Problem 4 Let hn] be the sequence whose Fourier transform H(w) is real and as follows and let g[n] = (-1)"h[n] a-3 pts) Plot G(w) for w E-π, π]. Detail your derivations. Make sure to show the maximuin value of G(w) b - [2 pts| Derive explicitly the impulse response of the following system n] Hint: Besides some graphical consideration, there is no calculation. The answer is mostly based orn the use of properties. c - 3 pts] Up to...
New problems for 2020 1. A topological space is called a T3.space if it is a T, space and for every pair («,F), where € X and F(carefull), there is a continuous function 9 :X (0,1 such that f(x) 0 and f =1 on F. Prove that such a space has the Hausdorff Separation Property. (Hint: One point subsets are closed.] 2. Let X be topological space, and assume that both V and W are subbases for the topology. Show...
Sur I Nano 2019/20) Qunntum physics exercices 1.1 Linear algebra and formalism Exercice 1.1.1 Basic calculations We consideran llibert prace Ey of dimension 2 and the two following vectors of En = (17.) and le >= () acting on vectors of Eh 21- We consider also the linear operator = 1+5 1 Calculate the square norms of the two vectors < Hermitian scalar products <ul> and < > 2. Calculate the eigensalues and cigarvectors of A. >, < > and...
Please only solve part C
Assume the following state space representation of a discrete-time servomotor system. (As a review for the Final Exam, you might check this state space representation with the difference equation in Problem 1 on Homework 2. This parenthetical comment is not a required part for Homework 8.) 2. 0.048371 u(n) 1.9048x(n) lo.04679 [1,0]x(n) y(n) Compute the open-loop eigenvalues of the system. That is, find the eigenvalues of Ф. Check controllability of the system. Or, answer the...
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