Consider the standard basis (e1, e2) of the two-dimensional Minkowski space (R1+1,m) with respect...
Consider the canonical basis (e1, e2) of the two-dimensional Cartesian space. Rotate the basis by an angle a counterclockwise. Show that the original cartesian variables (x1, x2) are related to the rotated variables (y1, y2) according to x = Ay, or equivalently, Show that A is orthogonal.
Consider the null cone of the three-dimensional Minkowski space (R2+1,m A. Write the equation of N in standard coordinates (t,,2) of R2+1 B. Let p (a,b,c) be a point (not the origin) on M. Draw the tangential plane to N at p. Moreover, draw all null vectors with origin at p. Consider the null cone of the three-dimensional Minkowski space (R2+1,m A. Write the equation of N in standard coordinates (t,,2) of R2+1 B. Let p (a,b,c) be a point...
7. [2p] (a) In a two-dimensional linear space X vectors el, e2 formi a basis. In this basis a vector r E X has expansion x = 2e1 + e2. Find expansion of the vector x in another basis 1 -2 er, e2, of X, if the change of basis matrix from the basis e to the basis e, s (b) In a two-dimensional linear space X vectors el, e2 forn a basis. In this basis a vector r E...
We equip the vector space R2 with a (non-standard) inner product (with respect to the standard basis E- {(1,0)", (0, 17) is ), whose metric Let L: R2 → R2 be the reflection operator with respect to the x-axis, defined by 21 21 Compute the adjoint operator Lt. Is L self-adjoint? We equip the vector space R2 with a (non-standard) inner product (with respect to the standard basis E- {(1,0)", (0, 17) is ), whose metric Let L: R2 →...
1 point) Read 'Diagonalization Changing to a Basis of Eigenvectors' before attempting this problem. Suppose that V is a 5-dimensional vector space. Let S -(vi,... , vs) be some ordered basis of V, and let T-(wi.... . ws) be some other ordered basis of V. Let L: V → V be a linear transformation. Let M be the matrix of L in the basis Sand et N be the matrix of L in the basis T. Decide whether each of...
4. Consider the vector space V = R3 and the matrix 2 -1 -1 2 -1 -1 0 2 We can define an inner product on V by (v, w) = v'Mw. where vt indicates the transpose. Please note this is NOT the standard dot product. It is a inner product different (a) (5 points) Apply the Gram-Schmidt process to the basis E = {e1,e2, e3} (the standard basis) to find an orthogonal basis B. 4. Consider the vector space...
3. An almost-flat spacetime has metric in coordinates (zo, х, x2,T3) = (ct, x, y, z), where ηοβ = diag(-1, 1,1 , 1) is the Minkowski metric and the perturbation has is small, with hasl1. Let hue huv Σημι,h, where h y®ßhaß. The Einstein field equation becomes in the absence of matter and omitting terms of order ha. Consider a change of coordinates frorn zor to r/a-r" + ξα, in which the functions ξα are comparable in size to the...
APM 346 (Summer 2019), Homework 1. 5. Consider the two-dimensional vector space of functions on the interval [0, 1 V = {a sin mz + bcos π.rla, b e R). (a) Prove that B is a basis for V. (Hint: Wronskian!) (b) Find the matrix representation [T]B of the operator T in the basis B, for (i) T = 4; (ii) T = ar . APM 346 (Summer 2019), Homework 1. 5. Consider the two-dimensional vector space of functions on...
Consider a two-dimensional state vector space and a basis in this space lay), laz), eigenvectors of an observable A: A|az) = ajlaj) Alaz) = azlaz) A representation of Hamiltonian operator in this basis is: h = (9) - Calculate the dispersion of measurements of A and Hat time t. Is the uncertainty principle Energy x Time fulfilled?
Consider a two-dimensional state vector space and a basis in this space lay), laz), eigenvectors of an observable A: Ala) = aja) Alaz) = azlaz) A representation of Hamiltonian operator in this basis is: H = (8 5) Find: -Eigenstates and eigenvalues of H. -If the system is in state |az) at time t=0, What is the state vector of the system at time t? -What is the probability of finding the system in the state |az) at time t?...