Reu Apprentice Problems Instructors: Madhur Tulsiani and László Babai

REU 2013 : Apprentice Program Summer 2013 Lecture 15 : July 22 , 2013

Apprentice Linear Algebra , 4 th lecture , 07 / 08 / 05 REU 2005 Instructor : László Babai Scribe :

a system has a solution. Recall that if x =  x1 .. x` , and the matrix A is written as A = [ a1 . . . a` ] , where ai is the ith column of the matrix, then Ax = ∑i=` i=1 xiai is simply a linear combination of the columns. Therefore if Ax = b holds, then b ∈ Span{a1, . . . ,a`}. Recall that the span of the ai is called the column space of the matrix A. It follows that rk(A) = rk ([A|b]), whe...

Chicago Math REU Apprentice Program Exercises Friday , July 24 Instructor : László Babai Scribe : Angela

Exercise 25.2. Recall that for A ∈Mn(R) with A = A , the Rayleigh quotient is defined as RA(x) = xTAx xT x . For symmetric φ : V → V , the Rayleigh quotient is defined as Rφ(x) = 〈x,φx〉 ||x||2 . Use the above theorem to show that argmaxxRφ(x) is achieved on V \ {0}. Proof. Note that Rφ(αx) = Rφ(x). So any value achieved by Rφ on V \ {0} is achieved on the unit sphere {x ∈ V | ||x|| = 1}. Restri...

Apprentice Linear Algebra , 3 rd day , 07 / 06 / 05 REU 2005 Instructor : László Babai

Regarding the entire discussion of vector spaces and their properties, we may replace R by any field F , and define vector space “over F” for which F is the domain of scalars. Note that it only makes sense to consider linear maps between vector spaces over the same field. We are already familiar with the field of real numbers R, but also with rational numbers Q, complex numbers C, and the finit...

REU Apprentice Monday , fourth week