4.12 For (u, v) F, symplectic weight wts((u, v)) of (u, v) is defined to be the number of 1sin such that at least on...
QUESTION 8 Let V = U ㊥ W where V is a finite-dimensional vector space over a field F, and U and w are subspaces of V. Suppose U1 and U2 are subspaces of U and Wi and W2 are subspaces of W Show that QUESTION 8 Let V = U ㊥ W where V is a finite-dimensional vector space over a field F, and U and w are subspaces of V. Suppose U1 and U2 are subspaces of U...
4.11 Let )s F 2n F2n F be defined as (u, v), (u', v)s u .v -vu where u, v, u', v' e F and is the Euclidean inner product on F. Show that )s is an inner product on F. (Note: this inner product is called the symplectic inner product. It is useful in the construction of quantum error-correcting codes.) 4.11 Let )s F 2n F2n F be defined as (u, v), (u', v)s u .v -vu where u,...
(3) Consider f: R3- R3 defined by (u,, w)-f(r, y, :) where u=x w = 3~. Let A = {1 < x < 2, 0 < xy < 2, 0 < z < 1). Write down (i) the derivative Df as a matrix (ii) the Jacobian determinant, (ii) sketch A in (x, y. :)-space, and iv) sketch f(A) in (u. v, w)-space.
Unemployment. Recall the model of long-run unemployment: U' (1-f) . U + s , E, = where E denotes current employment, U denotes current unemployment, s denotes the sep- aration rate, f denotes the job finding rate, and E and U" denote future employment and unemployment. As usual, we define the labor force as: L= E + U and the unemployment rate as: Answer the following (a) (3 points) Derive an expression for the steady state or natural rate of...
3. (Section 11.3) Explain using 1-2 sentences why u + v.w is not defined, where u, v, w are all nonzero vectors. Hint: think of the difference between a scalar and a vector, as well as what type of answer you get when computing a dot product.
Asvanced Calculus 12. Consider A = R'. Ifu, v E A, the Hamming distance is defined by d(u, v) to be the number of coordinates in which they differ. For example if u = (0,1,2) and v = (0,5,6) then d(u, v) = 2 since the vectors differ in the 2nd and 3rd coordinate, but agree in the 1st. (a) Show that d(u, v) is a metric on A. (b) Let S be the subset of A consisting of the...
Unemployment. Recall the model of long-run unemployment E' f-U+(1-8) . E = where E denotes current employment, U denotes current unemployment, s denotes the se aration rate, f denotes the job finding rate, and E and U denote future employment and unemployment. As usual, we define the labor force as and the unemployment rate as: Answer the following (a) (3 points) Derive an expression for the steady state or natural rate of unemployment u" as a function of the job...
Please answer A, B, and C in full 2. Let f() € F[2] be a separable polynomial with roots {u1, ..., Un} contained in some splitting field K of f(x) over F. Define A= || (ui-u) = (ui - U2) (u - u3) ...(ui-un)(uz - u3) ..(un-1 - Un) EK. (a) (15 points) Consider GalpK < Sn by looking at its action on the set of roots for f(x). Show that if Te Galo K is a transposition then (A)...
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for all x € R2,AG R, but is not a linear transformation. (b) Show that a linear transformation VWfrom a one dimensional vector space V is com- pletely determined by a scalar A (e) Let V-UUbe a direet sum of the vector subspaces U and Ug and, U2 be two linear transformations. Show that V → W defined by f(m + u2)-f1(ul) + f2(u2) is...
How does one solve this problem? 4. (a) Consider the vector space consisting of vectors where the components are complex numbers. If u = (u1, u2, u3) and v = (V1,V2, us) are two vectors in C3, show that where vi denotes the complex conjugate of vi, defines a Hermitian (compler) inner product on C3, i.e. 1· 2· 3, 4, (u, v) = (v, u), (u+ v, w)=(u, w)+(v, w), (cu, v) = c(u, v), where c E C is...