Ex12. Let L:R5 → R4 be defined byXy(EDL1 0-110 020-10 0 -1 1-3252-1-1-10XyZVa) Find ker L, basis for ker L, dim(ker L) anddetermine if L one to one.b) Find range L, basis for range L, dim(range L)and determine if Lonto.

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Answered: 4 Ex12. Let L:R5 → R* be defined by X y…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2. Suppose that z is a function of r and y, and F(xz - y², xy - 2², yz - ²) = 0. Calculate and (Hint: Since F = 0, then dF = 0. Use a proper change of variables and chain rule to proceed)
Let T: R³ R2 be a linear mapping. → 10 *([:]) = [ ] ¹ ({]) - [ ] -~-~(:)) - [²7] ([]) - T = and T find T (ED). Given that T sin (a) Ər f α Ω E
14.4 , 3
5. For n, m >0 and n + m, show T„(r)T (1) dr 0 VI- V1-r This is called the orthogonality relation for Chebyshev polynomials. Hint: Use T,(r) = cos(n cos-(x)) and z = cos e.
4.7.6. For the operator T: R² → R² defined by T(x, y) = (x+y, –2x +4y), {(}). (;)}- determine [T]B, where B is the basis B =
2. Let P denote the path that goes in straight lines from (1,1,1) to (-1,-1,1) to (-1,-1,-1) to (1,1,-1) and then back to (1,1, 1). 2xyz+z el +2 x²z+sin(z) Compute fp F. dr, where F(x, y, z) = P
Ques. 3: Check whether the operator T: R2 → R? defined by T(x, y) = (y,x) is linear. Moreover if T is linear find T' if exists.
Exercise 4.3.4. (1) Find two functions h,k: R → R such that neither h nor k is a constant map, but ko h is a constant map. (2) Find two functions s,t: R → R such that s‡ 1r and t ‡ 1r, but tos = 1R. R
from ( 0,1) to (2,5)**

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4 Ex12. Let L:R5 → R* be defined by X y EH = W L -1 3 02 -1 5 0 0 -1 1 1 0 1 0 2 0 -1 -1 0 X y Z W Find ker L, basis for ker L, dim(ker L) and determine if L one to one. Find range L, basis for range L, dim(range I and determine if L onto.