11. (8 marks) Let F(x, y, z) = x'yz, where r, y,z E R and y, z 2 0. Execute the following steps to prove that F(z,y,2) < (y 11(a) Assume each of r, y, z is non-zero and so ryz= s, where s> 0. Prove that 3 F(e.y.) (y)( su, y su, z sw and refer back to Question (Hint: Set 10.) 11(b) Show that if r 0 or y0 or z 0, then F(z, y, z) ( 11(c)...
Define where S is the collection of all real valued sequences i.e. S = {x : N → R} and we denote xi for the ith element a the sequence x E S. Take for any x EL (i) Show that lic 12 (where recall 1-(x є s i Izel < oo)) (ii) Is l? Prove this or find a counterexample to show that these two sets do not coinside (iii) ls e c loc where recall looー(x є sl...
12. (True/False) (a) Let AE Rm*n . Then R(A) (b) Let AERm*n. Then N(A) is isomorphic to N(AT) (c) We define < A. B > = Tr (BTA ) where A, B E Rnxn . is isomorphic to R(A Then 〈 . , . 〉 is an inner product on Rmxn. (d) Consider a periodic-function space V with period of 1 sec. Define an inner product on V by <f,a>= f(t )a (t ) dt. Then cos 2 π t...
11. For xe R' and y e R', define d,(x, y) = (x-y)2, Determine, for each of these, whether it is a metric or not.
Let B C R" be any set. Define C = {x € R" | d(x,y) < 1 for some y E B) Show that C is open.
Let βˆ = (X′X)−1X′y where y ∼
N(Xβ,σ2I), X is an n×(k+1) matrix, and β is a (k+1)×1 vector. Are
βˆ′A′[A(X′X)−1A′]−1Aβˆ and y′[I − X(X′X)−1X′]y independent?
Let B (X'X)-X'y where y ~ N(XB,02I), X is an n x (k+ 1) matrix, and B is a (k+1) x1 vector Are BA A (X'X)-A]-AB and yI - X(X'X)-xy independent?
Let B (X'X)-X'y where y ~ N(XB,02I), X is an n x (k+ 1) matrix, and B is a (k+1) x1 vector Are...
8. Let V = {(x,y)x,y e R}. Define addition on V as follows: (x,x)+(x2,)=(x, +x,-1,, +y,+3) [4 marks] a. Prove addition axiom #3 (Addition is commutative). b. Find the zero vector.
Let x, y e R" for n e N, writing x-(n, ,%), similarly for y. The Euclidean Metric of $2.2 is often called the 12 metric and written |x - y l2 for x,y e R. Show that the following three similar relations are also metrics: (a) the tancab, or 11 metric: lx-wi : = Σ Iri-Vil (b) the marirnum, or lo-metric. Ilx-ylloo:=max(zi-yil c) the comparison, or 10-metric. Ix_ylo rn (ri-Vi) where δ(t)- if t = 0
Let y = Xß + ε where ε ~ N(0, σ21). Let β = (XTX)-"XTy and let è-y-X β. (a) Show that è-(1-Pxje where Px (b) Compute Ee -e 2. X(XTX)-1x" (1 (c) Compute Varle-e.
Let y = Xß + ε where ε ~ N(0, σ21). Let β = (XTX)-"XTy and let è-y-X β. (a) Show that è-(1-Pxje where Px (b) Compute Ee -e 2. X(XTX)-1x" (1 (c) Compute Varle-e.
Define a relation R from R to R as follows: For all (x, y) E R x R, (x, y) E R if, and only if, x= y2 + 1. (a) Is (2, 5) E R? Is (5, 2) e R? Is (-3) R 10? Is 10 R (-3)? (b) Draw the graph of R in the Cartesian plane. (C) Is R a function from R to R? Explain.