MATH311ansHW7

# MATH311ansHW7 - Math 311 Homework 7 partial answers and solutions 3.4.5 f R 3 → R 2 is a linear function and has a matrix representation f x y z

This preview shows pages 1–3. 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 311, Homework 7 partial answers and solutions 3.4.5. f : R 3 → R 2 is a linear function, and has a matrix representation f x y z = 1 4 3 2 5 4 x y z . So its image is the span Span 1 2 , 4 5 , 3 4 Since 1 = 3 4- 2 1 2 and 1 = 1 2 + 3 4- 4 5 , this span is in fact all of R 2 , so the function is onto. Finally, to find its null space we solve 1 4 3 2 5 4 x y z = . Using row reduction this is equivalent to 1 4 3- 3- 2 x y z = , whose solutions are x y z = 8 t- 9 t- 2 t 3 t = Span - 1- 2 3 . 3.4.6. Answer: R 3 , not a subspace, not linear. 3.4.9. The image consists of all functions g ( x ) in C (1) (-∞ , ∞ ) with g (0) = 0 . Indeed, clearly any g = F ( u ) has to satisfy this condition. Conversely, take any g like this, and let u ( x ) = e x g ( x ) . Then u is in C ( ∞ , ∞ ) and F ( u )( x ) = g ( x ) . The function is linear. If F ( u )( x ) = 0 for all x , then F ( u ) ( x ) = 0 for all x . But this means that e- x u ( x ) = 0 for all x , so that u ( x ) = 0 for all x , and the null-space of F contains only the zero function. 1 3.4.13. The image is Span 2 2 , 4 1 1 , 1 1 = Span 4 1 1 , 1 1 . The null-space consists of all the solutions of 2 4 1 0 1 0 2 1 1 x y z = , which is equivalent to 2 4 1 0 1 0 0 0 0...
View Full Document

## This note was uploaded on 04/29/2008 for the course MATH 311 taught by Professor Anshelvich during the Spring '08 term at Texas A&M.

### Page1 / 5

MATH311ansHW7 - Math 311 Homework 7 partial answers and solutions 3.4.5 f R 3 → R 2 is a linear function and has a matrix representation f x y z

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online