Consider the canonical basis (e1, e2) of the two-dimensional Cartesian space
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.
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Consider the canonical basis (e1, e2) of the two-dimensional Cartesian space
Consider the standard basis (e1, e2) of the two-dimensional Minkowski space (R1+1,m) with respect...
Consider the standard basis (e1, e2) of the two-dimensional Minkowski space (R1+1,m) with respect to which the metric tensor m takes the form -1 0 Consider the new basis Compute the metric matrix m with respect to the basis (L, L)
Consider the standard basis (e1, e2) of the two-dimensional Minkowski space (R1+1,m) with respect to which the metric tensor m takes the form -1 0 Consider the new basis Compute the metric matrix m with respect to the basis...
Consider the following equations: y1 = a1y2 +a2y3 +x1 +x2 +e1 (1) y2 = by1+2x3+x1+e2 (2) y3 =cy1+e3 (3) Here a1, a2, b, c are unknown parameters of interest, which are all posi- tive. x1, x2, x3 are e...
Consider the following equations: y1 = a1y2 +a2y3 +x1 +x2 +e1 (1) y2 = by1+2x3+x1+e2 (2) y3 =cy1+e3 (3) Here a1, a2, b, c are unknown parameters of interest, which are all posi- tive. x1, x2, x3 are exogenous variables (uncorrelated with y1, y2 or y3). e1, e2, e3 are error terms. (a) In equation (1), why y2,y3 are endogenous? (b) what is (are) the instrumental variable(s) for y2, y3 in equation (1)? (no need to explain why) (c) In...
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 expansio...
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...
Consider a two-dimensional state vector space and a basis in this space lay), laz), eigenvectors of...
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...
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?...
please help me with questions 1，2，3 1. Let V be a 2-dimensional vector space with basis...
please help me with questions 1，2，3
1. Let V be a 2-dimensional vector space with basis X = {v1, v2}, write down the matrices [0]xx and [id]xx. 2. Let U, V, W be vector spaces and S:U +V, T:V + W be linear transforma- tions. Define the composition TOS:U + W by To S(u) = T(S(u)) for all u in U. a. Show that ToS is a linear transformation. b. Now suppose U is 1-dimensional with basis X {41}, V...
Classical Mechanics Let us consider the following kinetic (T) and potential (U) energies of a two-dimensional...
Classical Mechanics Let us consider the following kinetic (T) and potential (U) energies of a two-dimensional oscillator : ?(?,̇ ?̇)= ?/2 (?̇²+ ?̇²) ?(?,?)= ?/2 (?²+?² )+??? where x and y denote, respectively, the cartesian displacements of the oscillator; ?̇= ??/?? and ?̇= ??/?? the time derivatives of the displacements; m the mass of the oscillator; K the stiffness constant of the oscillator; A is the coupling constant. 1) Using the following coordinate transformations, ?= 1/√2 (?+?) ?= 1/√2 (?−?)...
Write a MATLAB Graphical User Interface (GUI) to simulate and plot the projectile motion – the...
Write a MATLAB Graphical User Interface (GUI) to simulate and plot the projectile motion – the motion of an object projected into the air at an angle. The object flies in the air until the projectile returns to the horizontal axis (x-axis), where y=0. This MATLAB program should allow the user to try to hit a 2-m diameter target on the x-axis (y=0) by varying conditions, including the lunch direction, the speed of the lunch, the projectile’s size, and the...
A group of physics students collected data from a test of the projectile motion problem that...
A group of physics students collected data from a test of the projectile motion problem that was analyzed in a previous lab exercise (L5). In their test, the students varied the angle and initial velocity Vo at which the projectile was launched, and then measured the resulting time of flight (tright). Note that tright was the dependent variable, while and Vo were independent variables. The results are listed below. (degrees) Time of Flight (s) Initial Velocity V. (m/s) 15 20...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...
Consider the standard basis (e1, e2) of the two-dimensional Minkowski space (R1+1,m) with respect to which the metric tensor m takes the form -1 0 Consider the new basis Compute the metric matrix m with respect to the basis (L, L)
Consider the standard basis (e1, e2) of the two-dimensional Minkowski space (R1+1,m) with respect to which the metric tensor m takes the form -1 0 Consider the new basis Compute the metric matrix m with respect to the basis...
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...
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?...
please help me with questions 1，2，3
1. Let V be a 2-dimensional vector space with basis X = {v1, v2}, write down the matrices [0]xx and [id]xx. 2. Let U, V, W be vector spaces and S:U +V, T:V + W be linear transforma- tions. Define the composition TOS:U + W by To S(u) = T(S(u)) for all u in U. a. Show that ToS is a linear transformation. b. Now suppose U is 1-dimensional with basis X {41}, V...
A group of physics students collected data from a test of the projectile motion problem that was analyzed in a previous lab exercise (L5). In their test, the students varied the angle and initial velocity Vo at which the projectile was launched, and then measured the resulting time of flight (tright). Note that tright was the dependent variable, while and Vo were independent variables. The results are listed below. (degrees) Time of Flight (s) Initial Velocity V. (m/s) 15 20...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...
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