T:A"-+A" if and Only ir- Show',that maxxeacT(x)lo is positive. (b) Conclude that the linear mapping T:is...
2. Let b(1,-1,1). Define T: R3R3 by the mapping: T(x) (x b)b (a) Show that T is a linear transformation by verifying the two linear transformation axioms (b) Determine the standard matrix representation for T. (c) Give a geometrical interpretation of T. 2. Let b(1,-1,1). Define T: R3R3 by the mapping: T(x) (x b)b (a) Show that T is a linear transformation by verifying the two linear transformation axioms (b) Determine the standard matrix representation for T. (c) Give a...
Let b-,-1,1). Define T:RR by the mapping: V3 T(x)-(x b)b (a) Show that T is a linear transformation by verifying the two linear transformation axioms. b) Determine the standard matrix representation for 1 (c) Give a geometrical interpretation of T
Show that Q defined on (C [0, 1], || . ||) by 3 (b) 1 Q(x) tx(t)dt is a bounded linear Show that Q defined on (C [0, 1], || . ||) by 3 (b) 1 Q(x) tx(t)dt is a bounded linear
PART C ONLY! Thank you. 14. Fix a non-zero vector n R". Lot L : Rn → Rn be the linear mapping defined by L()-2 proj(T), fa TER or all (a) Show that if R", Such that oandj-n -0, then is an eigenvector of L What is its cigenvaluc? (b) Show that is an cigenvector of L. What is its cigenvalue? (c) What are the algebraic and geometric multiplicities of all cigenvalues of L? 14. Fix a non-zero vector n...
an converges. 6. We want to use the Integral Test to show that the positive series All of the following need to be done except one. Which is the one we don't need to n=1 do? (a) Find a function f(x) defined on [1,00) such that f(x) > 0, f(x) is decreasing, and f(n) = an for all n. (b) Show that ſi f(x) de converges. (C) Show that lim f(x) dx exists. t-00 (d) Show that lim sn exists....
an a Show A function TR → (From IR" to com is called a linear transformation of i) T(V+0) = T(V) + T(U) i T(V) = KTV) for all V, UER", KER. Let A be mxn matrix. that T(V) = AV is linear transformation from Rh to som (ie show properties i, ii are true. Appeal to the properties of matrix multiplication Covered in lecture u Let A be a 2x2 matsix. This corresponds to a Imear transformation from LR2...
6. Let L be the linear operator mapping R3 into R3 defined by L(x) Ax, where A=12 0-2 and let 0 0 Find the transition matrix V corresponding to a change of basis from i,V2. vs) to e,e,es(standard basis for R3), and use it to determine the matrix B representing L with respect to (vi, V2. V
Use the well-ordering principle of natural numbers to show that for any positive rational number x ∈ Q, there exists a pair of integers a, b ∈ N such that x = a/b and the only common divisor of a and b is 1.
1. Given f(x) = et XER g(x) = 3lnx x > 0,x & IR a) Find an expression for fg(x), simplifying your answer [2] b) Show that there is only one real value of x for which fg(x) = gf(x) (6) 8 Marks
Show that a metric space X is not connected if and only if there exists a continuous surjection f : X → {0, 1}.