Problem 6 (Successive Transformations): Let (A), B), (C), (U) denote four frames. Given the trans...
Problem 5 Let U be an n dimensional vector space and T E L(U,U). Let I denote the identity transformation I(u) = u for each u EU and let 0 denote the zero transformation. Show that there is a natural number N, and constants C1, ..., CN+1 such that C1I + c2T + ... + CN+1TN = 0 (Hint: Given dim(U) = n, what is the dimension of L(U,U)? consider ciI + c2T + ... + Cn+11'" = 0, where...
Problem 6. A fair die is rolled four times. (a) Let Y denote the number of distinct rolls. Find the probability mass function of Y. (b) Let Z denote the minimal result fo the 4 throws. Find the probability mass function of Z
Let u= -3 2 4 ; and let L denote the line thru the origin of R3 in the direction of u. The projection of R3 onto L — denoted PL : R3 −→ R3 — is definded to be equal to the projection pu onto the vector u. You may assume that PL is a linear transformation. Find the standard matrix [PL] for PL.
Exercise 2: Möbius Transformations I (a) [10 points] Denote A := {z € C: |z| < 1}. Prove the following statement. Every Möbius transformation g: A → A who maps A onto A can be written as 9(2) = e® (2- 20 Zoz – 1 with 0 eR and |zo| < 1. Conversely, each such function maps A onto A. (b) [6 points] Find a Möbius transformation f with f(i) = i, f (0) = 0 and f(-i) = 0....
QUESTION 5 Let V denote an arbitrary finite-dimensional vector space with dimension n E N Let B = {bi, bn} and B' = { bị, b, } denote two bases for V and let PB-B, be the transition matrix from B to B' Prove that where 1 V → V is the identity transformation, i e 1(v) v for all v E V Note that I s a linear transformation 14]
QUESTION 5 Let V denote an arbitrary finite-dimensional vector...
Linear Algebra Problem!
1. Let U be the subspace of R3 given by 11 + 12 - 213 = 0. for U. Justify that is an ordered basis. What is the a) Find an ordered basis dimension of U? b) Let ū= (1,1,1). Show that ✓ EU and find the B-coordinate vector (Ū3 = C:(Ū), where Ce: U + R2 is the B-coordinate transformation.
Both question
Let T denote the linear transformation corresponding to the matrix B-A-A. Find T Hint: You do not have to calculate A2. (ld) (2 marks) Let S:R2 find s [2 R? denote a linear transformat tion such that SandThen
Question 19: Linear Transformations Let S = {(u, v): 0 <u<1,0 <v<1} be the unit square and let RCR be the parallelogram with vertices (0,0), (2, 2), (3,-1), (5,1). a. Find a linear transformation T:R2 + R2 such that T(S) = R and T(1,0) = (2, 2). What is T(0, 1)? T(0,1): 2= y= b. Use the change of variables theorem to fill in the appropriate information: 1(4,)dA= S. ° Sºf(T(u, v)|Jac(T)| dudv JA JO A= c. If f(x, y)...
please solve the Q1,Q2...
Throughout these exercises, A, B, and C denote orthogonal transformations or their matrices), and T is translation by a 1. Prove that CT, = TeaC. 2. Given isometries F = T,A and G = T,B, find the translation and orthog onal part of FG and GlF.
Throughout these exercises, A, B, and C denote orthogonal transformations or their matrices), and T is translation by a 1. Prove that CT, = TeaC. 2. Given isometries F =...
Determinants and linear transformations 4. (a) Let A be the matrix 1 -2 4 1 3 2 11 i) Calculate the determinant of A using cofactor expansion of row 3. (ii) Is A invertible? If so, give the third column of A1 (you do not have to simplify any fractions) (b) Let B be the matrix 0 0 4 0 2 8 0 4 2 1 0 0 0 7 Use row operations to find the determinant of B. Make...