601-hw-iii

# 601-hw-iii - MATH 601, AUTUMN 2008 Additional homework III,...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 601, AUTUMN 2008 Additional homework III, October 18 PROBLEMS 1. Determine dim T ( R 3 ) for the linear operator T : R 3 R 4 given by T x y z = 1- 1 1 2 6- 1 3 3 1 1 5 x y z , with vectors of R 3 and R 4 written in the column form. 2. For any integer n 3, find dim W , where W denotes the vector space of all polynomials f of the real variable x with deg f n and f (1) = f (0) = R 1 f ( x ) dx = 0. 3. Given an integer n 0, let W be the vector space of all polynomials f of the complex variable z with deg f n and f (0) = 0. Find det T , where T : W W is the linear operator given by ( Tf )( z ) = zf ( z ). 4. Let a unit of measure | | in a 4-dimensional real vector space V be chosen in such a way that | K | = 7, where K is the parallelepiped in V with the given edge vectors p, q,r,s V emanating from 0. Evaluate | K | for the parallelepiped K in V with the edge vectors u, v,w,z such that u = p- s , v = p + q + r , w = p- r and z = q + s . 5. Denoting | | the standard unit of measure of R 3 , find the volume | K | , where K is the image of the cube { ( x,y, z ) R 3 : x,y, z [0 , 1] } under the linear operator T : R 3 R 3 defined by T x y z = 2- 1 3 1 2 2 2 x y z , with vectors of R 3 written in the column form. 6. In the 3-dimensional real vector space V of all C 3 functions f : R R with f 000 = 0 everywhere, define the volume form by ( f,g,h ) = f (1) g (2) h (3)- f (1) g (3) h (2) + f (2) g (3) h (1)- f (2) g (1) h (3) + f (3)...
View Full Document

## This note was uploaded on 04/18/2009 for the course MATH 601 taught by Professor Un during the Fall '08 term at Ohio State.

### Page1 / 5

601-hw-iii - MATH 601, AUTUMN 2008 Additional homework III,...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online