Assume the mapping T: P, -* P, defined by Tín *** ++2,1") + sa, + (70,...
Assume the mapping T: Pz --P, delined by T (* *24t+azt") =90, * (34,-33,)+ (50,- 203) P is inear. Find the matrix representation of T relative to the basis 8 = {1,1,1) The matrix for relative to Bis
10. Let T : P P , be the linear transformation defined by T(P) = (a) What is the kernel of T? (b) According to the concept of the rank theorem, what is the dimension of the range of T? (C) (needs an idea from earlier in the semester) If we represent P, by coordinate vectors rela- tive to it's standard basis (1.1.1-.1') and P, by coordinate vectors relative to it's standard basis (1,1,1"), find the standard matrix A of...
PROBLEM 2.15] Consider the linear mapping T : R4 → R3, defined as T1 T2 | = 5x1 + 2x2 + 7x3 + 24 42:1 + 322 + 713-214 ) T4 (i) Write the corresponding matrix [T]. (i) Find a basis of Range(T).1 (ii Find a basis of Null(T).[1) (iv) Find the rank of T in 3 different ways.[1 (v) Show that T satisfies the rank theorem. 1
6. Let T: P, – P, be the linear operator defined as T(p(x)) = p(5x), and let B = {1,x,x?} be the standard basis for Pz. a.) (5 points) Find [7]s, the matrix for T relative to B.
(10) Let TEL(P3(C)) be defined by T(P(x)) = p” (x) – p(0), where the prime symbol denotes differentiation. (i) (5 marks) Let y = {x2 + 2x – 3, x, x3 – 1,1} be an ordered basis and ß the standard ordered basis for P3(C). Determine the matrix representation [T]3. (ii) (4 marks) Determine a basis for ker(T).
3. This example hopes to illustrate why the vector spaces the linear transformation are defined on are critical to the question of invertibility. Let L : → p, be defined by L(p)(t+1)p(t)-plt). (a) Given a basis of your choice, find a matrix representation of I with respect to your chosen basis (b) Show L: P+P is not invertible (e) Let V-span+21-4,+2t-8). It can be shown that L VV. Given an ordered basis for V of your choice, find a matrix...
6. Let T: P2P be the linear operator defined as T(p(x)) = P(5x), and let B = {1,x,x?} be the standard basis for P2 a.) (5 points) Find [T), the matrix for T relative to B. b.) (4 points) Let p(x) = x + 6x2 Determine [p(x)]s, then find T(p(x)) using [T]g from part a. c.) (1 point) Check your answer to part b by evaluating T(x + 6x) directly
(1 point) Let f:R → R'be the linear transformation defined by T 4 -5 51 f(T) = -1 2 - 5 . | -4 0 3 Let B = {(-2,-1, 1), (-2, -2,1),(-1,-1,0)}, C = {{-2, -1, 1), (2,0, -1),(-1,1,0)}, be two different bases for R3. Find the matrix f for f relative to the basis B in the domain and C in the codomain. IT 3
Let T be a linear map from R3[z] to R2[z] defined as (T p)(z) = p'(z). Find the matrix of T in the basis: 4 points] Let T be a linear map from Rals] to R12] defined as (TP)(z) = p,(z). Find the matrix of T in the basis: in R2[-]; ~ _ s, r2(z) (z-s)2 in R2 [2], where t and 8 are real numbers. T1(2 Find coordinates of Tp in the basis lo, 1, 12 (if p is...
How was the linear transformation of b1 and b2 were applied (L(b1) , L(b2))? NOTE: b1=(1,1)^T , b2=(-1,1)^T Linear Transformations EXAMPLE 4 Let L be a linear transformation mapping R? into itself and defined by where (bi, b2] is the ordered basis defined in Example 3. Find the matrix A represent- ing L with respect to [bi, b2l Solution Thus, A0 2 onofosmation D defined by D(n n' maps P into P, Given the ordered Linear Transformations EXAMPLE 4 Let...