MATH 2331 Lecture : Notes 2-03-2014.pdf

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peachrat235

12 Feb 2014

School

Northeastern University

Department

Mathematics

Course

MATH 2331

Professor

Rita Jimenez Rolland

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Document Summary

Examples: a. ) exp: r r x ex b. ) T : r3 r3 (cid:126)x orthogonal projection onto the xy-plane; t. , y r : x1, x2 r s. t. Y1 (cid:19)(cid:18)x1 (cid:18)1 3 (cid:19) (cid:18) x1 + 3x2 (cid:19) (cid:18)1 (cid:19) T ((cid:126)x) = (cid:19) x2 x3 x2 x3. 1 3 x2 x1 + 2x2 x1 + 3x2. 1 c. ) im(t ) = {t ((cid:126)x) : (cid:126)x r2}; set of all linear combinations of x2 x3. * im(t) is the set of all linear combinations of the columns of the transformation matrix a. The span of vectors (cid:126)v1, (cid:126)v2, , (cid:126)vk rm is de ned as follows: span((cid:126)v1, (cid:126)v2, , (cid:126)vk) = { 1(cid:126)v1 + 2(cid:126)v2 + + k(cid:126)vk : 1k r} T ((cid:126)x) = a(cid:126)x im(t ) = span.

Related Questions

Let P denote the vector space of all polynomials and f'(x)denote the derivative of all numbers in P. State true or false tothe following:

1. T: R^2 --> R^3 defined by T(x,y)=(?^2x,0, y/2) is a linear transformation

2. T: R^2 --> R^3 defined by T(x,y)=(x+y, x-y, xy) is alinear transformation

3. T: R^2 --> R^3 defined by T(x,y)=(1,0,0) is a lineartransformation

4. T: R^2 --> R^3 defined by T(x,y)= (y, x, y) is a lineartransformation

5. T: R^2 --> R^3 defined by T(x,y)= (x, x^2, x^3) is alinear transformation

6. T: R^2 --> R^3 defined by T(x,y)= (2x+3y, 3x+4y, 4x+5y) isa linear transformation

7. T: P --> P defined by T(f(x))= f(0) +f'(0)x + f''(0)x^2 isa linear transformation

8. T: P --> P defined by T(f(x))= f(1) + f'(2) + f''(3)x^2 isa linear transformation

9. T: P --> P defined by T(f(x))= f(x-3) is alinear transformation

10. T: P --> P defined by T(f(x))= f(x)-3 is a lineartransformation

tantoad80

coppershark262

1) Show that the follwing limit does not exist

lim_(x,y)-(0,0)(xy+x²)/(x^²+y²)

2) Fin all the second orderpartial derivatives of

f(x,y,z)=x²tan^-1(y+z)

3) Find the equation of thetangent plane to the surface

z=x²-4y² atthe piont (1,2,-15).

4) Fin the linear approximation tothe function

f(x,y,z)=e^-2xy+cos(2yz) at the point (1,-1,2)

5) Fin the normal vector to thesurface

z²=x²-y² at the point (5,3,4)

6) You are standing on a surfacegiven by the equation

z=x^2-2y². Ifyou are standing at the poin (2,1,0), in which directio is thesteepest slope?

7) The temperature in the solarsystem is given by

T(x,y,z)=10⁵/(x²+y²+z²). If acomet travels along the path r(t)=(t, t²-4, 2t), howfast is the temperatur changing when t=2?

8) Fin the critical point of thefunction f(x,y)= e^4x-e^-y. Use the secondderivative test to classifyf them, if possible.

9) Fin the extreme values off(x,y)=4x2+3y2 on the square 0 is less than or equal x,y less thanor equal to 1.

ceruleancockroach23

MATH 2331 Lecture : Notes 2-03-2014.pdf

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