A square matrix E∈Mn×n(R) is idempotent if E2=E. It is symmetric if E = tE. (a) Let V⊆Rn be a subspace of Rn, and consider the orthogonal projection projV:Rn→Rn onto V. Show that the representing matrix E = [projV]EE of proj V relative to the standard basis E of Rn is both idempotent and symmetric. (b) Let E∈Mn×n(R) be a matrix that is both idempotent and symmetric. Show that there is a subspace V⊆Rn such that E= [projV]EE. [Hint: What...
(0) is a lower- Consider the matrix equation Lx u, where L triangular square matrix and x = (p" and u = (u)' are column vectors. In view of Example 97: Solve the n equations for the n variables x1,x2, . . . , rn respectively. 1-12, . Example 97 We can find general formulas that characterize the procedure used in the previous example. Suppose we want to solve the equation Ux = v, where x = (x)' and v-(v)'...
Let V be Rn with a basis B={b1,. bn); let W be Rn with the standard basis, denoted here by E and consider the identity transformation I VW, where l(x) x. Find the matrix for I relative to s and E. What was this matrix called in Section?
Consider the linear transformation T: Rn → Rn whose matrix A relative to the standard basis is given. A 2 2 (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) (A1, A2) -1 5 (b) Find a basis for each of the corresponding eigenspaces (c) Find the matrix A' for Trelative to the basis B', where B' is made up of the basis vectors found in part (b)
5 points 1. True of False: a. if A is an n x1 matrix and B is a 1 xn matrix, then AB is an n xn matrix. b. if A is an n x1 matrix and B is a 1 x n matrix, then BA is not defined. 20 points 2. Use the Invertible Matrix Theorem to determine which of the matrices below are invert- ible. Use as few calculations as possible. Justify your answers. [34 01 4 5...
Let A be an m × n matrix, let x Rn and let 0 be the zero vector in Rm. (a) Let u, v є Rn be any two solutions of Ax 0, and let c E R. Use the properties of matrix-vector multiplication to show that u+v and cu are also solutions of Ax O. (b) Extend the result of (a) to show that the linear combination cu + dv is a solution of Ax 0 for any c,d...
Consider a sequence of random variables X1, . . . , Xn, . . .where for each n, Xn ∼ t distribution. Apply Slutsky’s Theorem to show that as the degrees of freedom go to infinity, the distribution converges to a standard normal. (a) Let V1, . . . , V_n, . . . be such that Vn ∼ Chi Sq, n df. Find the value b such that V/n in probability −→ b. (b) Letting U ∼ N(0, 1),...
Please show all work in READ-ABLE way. Thank you so much in advance. Problem 2.2 n and let X ε Rnxp be a full-rank matrix, and Assume p Note that H is a square n × n matrix. This problem is devoted to understanding the properties H Any matrix that satisfies conditions in (a) is an orthogonal projection matriz. In this problem, we will verify this directly for the H given in (1). Let V - Im(X). (b) Show that...
4. Consider solving the linear system Ax = b, where A is an rn x n matrix with m < n (under- determined case), by minimizing lle subject to Ar-b. (a) Show that if A Rmxn is full (row) rank, where m n, then AA is invertible. Then show that r.-A7(AAT)-ibis a solution to Ax = b. (b) Along with part (a) and the solution aAT(AA)-b, show that l thus, z is the optimal solution to the minimization problem. and...
The molar volume in cm^3/mol of a binary liquid mixture at T and P is given by:V~ = 120 x1 + 70 x2 + (15 x1 + 8 x2) x1 x2a.) Find expressions for the partial molar volumes of species 1 and 2 at T and P.b.) Show that when these expressions are combined in accord with Eqn 11.11 the given equation for V~ is recovered.c.) Show that these expressions satisfy Eqn 11.14, the Gibbs-Duhem equation.d.) Show that at constant...