115apracticeexam1-sol

115apracticeexam1-sol - Practice Hour Exam #1 Solutions...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Practice Hour Exam #1 Solutions Math 115A Section 3 NAME: SOLUTIONS SCORES: 1. / 20 2. / 20 3. / 10 4. / 10 5. / 20 6. / 20 Total: / 100
Background image of page 1

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

View Full DocumentRight Arrow Icon
Problem 1 (20 points - 5 points each) (a) State what it means for a subset S of a vector space V to be linearly independent . S is linearly independent if whenever v 1 ,...,v n S are distinct and a 1 ,...,a n are scalars such that a 1 v 1 + ··· a n v n = ~ 0, then a 1 = ··· = a n = 0. (b) Define the dimension of a finite-dimensional vector space. The dimension of a finite-dimensional vector space V is the number of vectors in any basis for V . (c) Suppose T : V W is a linear transformation. Define the null space of T . The null space of T is the set N ( T ) = { x V | T ( x ) = ~ 0 W } . (d) Suppose T : V W is a linear transformation. Define the rank of T . The rank of T is the dimension of the range of T , dim( R ( T )).
Background image of page 2
Problem 2 (20 points - 4 points each) For each of the following statements, determine if they are true or false. If they are true, prove them. If they are false, provide a counterexample . (a) dim( P n ( F )) = n . FALSE! Since { 1 ,x,. ..,x n } is a basis for P n ( F ), we have dim( P n ( F )) = n + 1. (b) 3 distinct vectors in R 4 must be linearly independent. FALSE! There are many ways to disprove this. The fastest way is to say that if the set contains (0 , 0 , 0 , 0), then it must be linearly dependent. Another less perverse example is the set { (1 , 0 , 0 , 0)
Background image of page 3

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

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/02/2012 for the course MATH 115A 262398211 taught by Professor Fuckhead during the Spring '10 term at UCLA.

Page1 / 9

115apracticeexam1-sol - Practice Hour Exam #1 Solutions...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online