3. Let A be a 3 x 3 real-valued matrix. Show there must be at least...
Let V = Cº(R) be the vector space of infinitely differentiable real valued functions on the real line. Let D: V → V be the differentiation operator, i.e. D(f(x)) = f'(x). Let Eq:V → V be the operator defined by Ea(f(x)) = eax f(x), where a is a real number. a) Show that E, is invertible with inverse E-a: b) Show that (D – a)E, = E,D and deduce that for n a positive integer, (D – a)" = E,D"...
)-( 1 (c) Let C be a real 3 x 3 matrix and b be a real 3-vector. The general solution to the matrix equation Cx=b is given by 2 2 =X3 + -4 2 for all XER Let 10 y = -6 8 (i) Let z be a real 3-vector. Find the solution set to the matrix equation Cz=0 (ii) Calculate M1, M2 ER such that 2 y = M1 ( 3 + H2 ·()--() 1 (iii) Express Cy...
Let V = R3[x] be the vector space of all polynomials with real coefficients and degress not exceeding 3. Let V-R3r] be the vector space of all polynomials with real coefficients and degress not exceeding 3. For 0Sn 3, define the maps dn p(x)HP(x) do where we adopt the convention thatp(x). Also define f V -V to be the linear map dro (a) Show that for O S n 3, T, is in the dual space V (b) LetTOs Show...
34. Let V be the subspace of the vector space of all real- valued continuous functions that has basis S = {e'. e-}. Show that V and Rare isomorphic.
2) Let CI0,1] be the vector space of all continuous real valued functions with domain [0,1J.Let (f.8)-Co)ds be the inner product in C10.11 where fand g are two functions in CI0,1. Answer the following questions for f(x)-x and g(x)-cos. a) Find 《f4) and i g I where l.l denotes the length induced by this inner product,Show your work b) Determine the scalar c so that f-cg is orthogonal to f.Show all your work.
4. Let f: X Y +R be any real valued function. Show that max min f(x,y) < min max f(x,y) REX YEY yey reX
Consider a real-valued function u(x, y), where x and y are real variables. For each way of defining u(x, y) below, determine whether there exists a real-valued function v(x, y) such that f(z) = u(x, y) + iv(x, y) is a function analytic in some domain D C C. If such a v(x, y) exists, find one such and determine the domain of analyticity D for f(z). If such a v(x, y) does not exist, prove that it does not...
2. Let A be an n x n real symmetric matrix or a complex normal matrix. Prove that tr(A) = X1 + ... + and tr(AⓇA) = 1212 + ... +14.12 where ....... An are the eigenvalues of A repeated with multiplicity (for example, if n = 3 and the eigenvalues of A are -3 and 7 but -3 has multiplicity 2 then 11 = -3, 12 = -3, and Az = 7). 3. Let A be an n x...
1. (20 points) Let X (Xi, X, Xs) be a real random vector, where X, are identically dis- tributed and independent (ii.d.) zero-mean Gaussian real random variables. Consider the random vector Y given by where A is a 3 x 3 real matrix and b is a 3 x 1 real vector. Justify all your answers. (a) Find the covariance matrix Cx of x. (b) Find the mean vector EY] of Y (c) Express the covariance matrix Cy of Y...
3. Let U E Rnxn be an orthogonal matrix, i.c., UTU = UUT-1. Show that for any vector x E Rn LXTL we have |lU 2 2. Thus the 2-norm of a vector does not change when it is multiplied by an orthogonal matrix. 3. Let U E Rnxn be an orthogonal matrix, i.c., UTU = UUT-1. Show that for any vector x E Rn LXTL we have |lU 2 2. Thus the 2-norm of a vector does not change...