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
Unformatted text preview: CALIFORNIA INSTITUTE OF TECHNOLOGY Department of Mathematics Math 1b; Solutions to Homework Set 2 Due: January 22, 2008 1. We will see later in the course that the solution space of m independent homogenous linear equations in n m variables is of dimension n m ; for the moment we prove this by ad hoc methods in the special cases in this problem. In HW1 we saw that in (1)(4), S is the subspace defined by one homogeneous linear equation, so we expect dim( S ) to be 3 1 = 2. We calculate that the following pairs of vectors are solutions to the respective equations: (1) e 2 ,e 3 ; (2) e 1 e 2 , e 3 ; (3) e 1 e 2 , e 1 e 3 ; (4) e 1 + e 2 , e 3 . Let P be the set consisting of the pair of vectors; visibly P is independent, so P is a basis for L ( P ) and hence dim( L ( P )) = 2. Thus it remains to show that S = L ( P ). As P S , L ( P ) S by Problem 2 on this HW. Thus 2 = dim( L ( P )) dim( S ) dim( V 3 ) = 3 , by Problem 3 on this HW, so dim( S ) = 2 or 3. Further in the first case dim() = 2 or 3....
View
Full
Document
This note was uploaded on 04/28/2010 for the course MATH 1B taught by Professor Aschbacher during the Winter '07 term at Caltech.
 Winter '07
 Aschbacher
 Linear Algebra, Algebra, Linear Equations, Equations

Click to edit the document details