sol1 - EE 221: Linear Systems HW #1 Solutions Sam Burden...

This preview shows page 1. Sign up to view the full content.

EE 221 : Linear Systems HW #1 Solutions — Sam Burden — Sep 10, 2010 1 Note these solutions are overly terse and leave out some details; in your solutions, you should strive for greater clarity and rigor. Exercise 1. function? yes; matrix multiplication well-deﬁned injective? no; f (0 , 0 , 1) = f (0 , 0 , 0) surjective? no; f ( a,b,c ) = ( a,b, 0) Exercise 2. 0: Suppose x : x + 0 = x,x + θ = x . Then 0 = 0 + θ = θ . 1: Suppose x : x · 1 = x,x · I = x . Then 1 = 1 · I = I . GL n is not closed under addition, I + ( - I ) = 0 6∈ GL n , hence not a ﬁeld. Note: GL 1 is commutative. Exercise 3. Note that each degree- k polynomial can be identiﬁed with exactly one vector in R k +1 : α k s k + ··· + α 0 ( α k ,...,α 0 ) . This pairing respects vector addition and scalar multiplication, hence it determines a vector space structure on polynomials. Therefore a natural basis is obtained from the natural basis on R k +1 , ± s k ,..., 1 ² , and the dimension is k + 1. Exercise
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 04/11/2011 for the course EE 221A taught by Professor Clairetomlin during the Fall '10 term at Berkeley.

Ask a homework question - tutors are online