# sa1 - spaces: (a) x 1-x 2 + 4 x 4 + 2 x 5-x 6 = 0 2 x 1-2 x...

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

Self-Assessment 1 – Math 5A, Winter 2008 Answer the following questions without looking in the book. If you do not feel comfortable doing this, read the corresponding sections in the book, and then solve the problems, again without looking in the book. 1. State the deﬁnition of a vector space. 2. Prove that the set V = { f : R R | f is continuous } is a vector space. 3. Given a matrix A R n × m , show that the set W = { x R m | Ax = 0 } is a vector space. 4. Consider a linear diﬀerential equation of the form y 00 + p ( t ) y 0 + q ( t ) y = 0 , (1) where p and q are given continuous functions deﬁned in R . Show that the set S = { y : R R | y is a solution of (1) } is a vector space of dimension 2. 5. Consider the set W = { ( x, y ) | x 2 + y 2 = 1 } ⊂ R 2 . Is W a vector space? Why? 6. Solve the linear following linear systems and determine their solution

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: spaces: (a) x 1-x 2 + 4 x 4 + 2 x 5-x 6 = 0 2 x 1-2 x 2 + x 3 + 2 x 4 + 4 x 5-x 6 = 0 1 2 (b) 2 x 1-2 x 2 + 4 x 3 + 2 x 4 = 0 2 x 1 + x 2 + 7 x 2 + 4 x 4 = 0 x 1-4 x 2-x 3 + 7 x 4 = 0 4 x 1-12 x 2-20 x 4 = 0 (c) 3 x 1 + 6 x 3 + 3 x 4 + 9 x 5 = 0 x 1 + 3 x 2-4 x 3-8 x 4 + 3 x 5 = 0 x 1-6 x 2 + 14 x 3 + 19 x 4 + 3 x 5 = 0 7. Determine the dimension and ﬁnd a basis for W = { ( x 1 , x 2 , x 3 , x 4 ) ∈ R 4 | x 1 + x 3 = 0 , x 2 = x 4 } . 8. Determine the dimension and ﬁnd a basis for W = { ( x 1 , x 2 , x 3 ) ∈ R 3 | x 1 + x 2 + x 3 = 0 } ....
View Full Document

## This note was uploaded on 12/26/2011 for the course MATH 5A taught by Professor Rickrugangye during the Fall '07 term at UCSB.

### Page1 / 2

sa1 - spaces: (a) x 1-x 2 + 4 x 4 + 2 x 5-x 6 = 0 2 x 1-2 x...

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

View Full Document
Ask a homework question - tutors are online