3. Let T: R2 R2 be defined by T =i) Find a basis for Null(T)ii) Find a basis for Range(T)<-x y-([]) = [ + ][]=[:]-x+ya- 16-X+Y0 0a+

Answered: 3. Let T : R² → R² be defined by T = X…

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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Let W(s, t) = F(u(s, t), v(s, t)), where F, u, and v are differentiable, and the following applies. u(3,-2) = -6 us(3,-2) = 4 u(3,-2) = -7 Fu(-6, -4) = 0 v(3,-2) = -4 Vs(3,-2) = 9 v(3,-2) = 1 F(-6, -4) = 7 Find We(3,-2) and We(3,-2). Ws(3,-2) = W+(3,-2) =
1. Let f(x, y) =x+2y. Find f(0, 1), f (2, –1), f(a, a), and f(a + h, b) – f(a, b). 2. Let f(x, y) = xy². Find the respective values of f (0, 1), f(-1,2), f(104, 10-2), f (a, a), f(a+h, b), and f(a, b+ k) – f (a, b). %3D 3. Let f(x, y) = 3x² – 2xy+ y. Find f(1, 1), f(-2, 3), ƒ(1/x, 1/y), p = [f(x+h,y) – f(x, y)]/h, and q = [f(x, y + k) – f(x, y)]/k. %3D SECTION 11.2 / PARTIAL DERIVATIVES WITH TWO VARIABLES 411 4. Let f(x, y) = x² + 2xy + y². (a) Find f(-1,2), f(a, a), and f (a + h, b) – f(a, b). (b) Prove that f(2x, 2y) = 2f(x, y), and that f(tx, ty) = tf(x, y) for all t.
Find the condition of c such that x² + 2y² +3z² + c(xy + yz) > 0 for all x, y, z & R with x² + y² + z² # 0.
Let z = x + iy where x, y € R. Show that z²+z² = 2 if and only if x² - y² = 1.
Suppose that T :C C is R-linear. Show that T is C-linear if and if T(iz) = iTz for all z.
2. Let f(x) = 3x – 7, find a linear function g for which f (g)= I and for which f(g (x)) =x for any number x the function g is: a. g:x →+7. 3 x+7 b. g:x→- 3 7-x c. g:x→ 3 x+7 d. g:x→- 3 x-7 e. g:x→ 3
Compute the discriminant D(x, y) of the function. f(x, y) = x³ + y² − 6x − 2y² + 1 (Express numbers in exact form. Use symbolic notation and fractions where needed.) D(x, y): = Which of these points are saddle points? (-√2, 1) (-√2,0) (√2, -1) (√2,0) (-√2, -1) (√2, 1) Which of these points are local minima? (-√√2, -1) (√2, -1) (√2, 1) (-√2,0) (-√2, 1) (√2,0)

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3. Let T : R² → R² be defined by T = X Y -x+y x-y