Math 2370 Fall 2008 Practice Problems V Problem 1: Let Ti : X X , i = 1,2,., n be linear maps and T = T1T2 .Tn . Show that if T is invertible then T1 , T2 ,., Tn are all invertible. Problem 2: Show th
Math 2370 Matrices and Linear Operators Solutions and Hints to Practice Problems 13 Bojana Peji c Here are the solutions to the first 5 questions, we will cover Problems 6-10 in recitation on Thursday
Math 2370 Matrices and Linear Operators Solutions and Hints to Practice Problems 6 Bojana Peji c Problem 1. There are n choices for p(1), n - 1 choices for p(2), . , 1 choice for f (n), which gives n(
Math 2370 Matrices and Linear Operators
Solutions and Hints to Practice Problems 9
Bojana Peji
c
Problem 1. Recall problem 4 (c) from Homework 8: If T L(V ) and
every vector in V is an eigenvector of
Math 2370 Matrices and Linear Operators Solutions and Hints to Practice Problems 1 Bojana Peji c Remark. This first solution set is unusually long. I often went into great detail, explaining along the
Math 2370 Matrices and Linear Operators Solutions and Hints to Homework Problems 4 Bojana Peji c Problem 1. This was done well in general. Problem 2. The statement is true. Proof: Let R. Since l is a
Math 2370 Matrices and Linear Operators Solutions and Hints to Practice Problems 5 Bojana Peji c Problem 1. We have seen in the recitation that if a linear map A from a finite dimensional vector space
Math 2370 Matrices and Linear Operators Solutions and Hints to Practice Problems 7 Bojana Peji c Problem 1. By induction on n. Suppose first n = 2. Then, A= A1 0 0 A2 = A1 0 0 I I 0 0 A2 .
Using the L
Math 2370 Matrices and Linear Operators Solutions and Hints to Homework Problems 8 Bojana Peji c Well done everyone - it was a pleasure to grade many nice solutions. The solutions below all come from
Math 2370 Matrices and Linear Operators Solutions and Hints to Practice Problems 12 Bojana Peji c Problem 1. Let x (Y + Z) . Then (x, y + z) = 0 for all y Y and z Z. In particular, if z = 0 Z, then (x
Math 2370 Matrices and Linear Operators Solutions and Hints to Homework Problems 11 Bojana Peji c The homework was done well in general. Here are some hints and solutions. Problem 1. In (a) and (b) we
Math 2370 Matrices and Linear Operators Solutions and Hints to Practice Problems 3 Bojana Peji c Problem 1. (a) Discussion. (This is not a part of the proof, but it is a part of the thinking process f
Math 2370 Fall 2008 Practice Problems I Problem 1: Let V be the set of all (ordered) pairs of real numbers. Let x = (a1 , a 2 ) and y = (b1 , b2 ) be elements of V and the addition and multiplication
Math 2370 Fall 2008 Practice Problems II Problem 1: Show that if vectors x1 , x2 ,., xn are linearly independent, so are vectors x1 x 2 , x 2 x3 ,., x n 1 x n , x n . Problem 2: Show that the linear s
Math 2370 Fall 2007 Practice Problems III Problem 1: Consider the quotient space obtained by reducing the space P of polynomials modulo subspace M of P. If M = Pm, the subspace of polynomials of degre
Math 2370 Fall 2008 Practice Problems VII Due October 24 as a HOMEWORK Problem 1: (a) Use the adjoint formula to compute the inverse of the following matrix: cos 0 - sin 0 1 0 sin 0 cos (b) Use Cramer
Math 2370 Fall 2008 Practice Problems VIII Problem 1: Suppose that T L(V , V ) where V is finite dimensional, is such that every subspace with dimension dim V - 1 is invariant under T. Show that T is
Math 2370 Fall 2008 Practice Problems X Problem 1: Show that if Y and Z are subspaces of a finite-dimensional Euclidean space then (Y + Z ) = Y Z . Problem 2: Show that if A is a linear map of a finit
Math 2370 Fall 2008 Practice Problems XI Problem 1: Which of the following matrices is unitarily similar to a diagonal matrix and why (or why not)? 1 1 i 1 1 1 1 1 i 0 1 1 1 1 0 2 Problem 2: Given the
Math 2370 Matrices and Linear Operators Solutions and Hints to Practice Problems 10 Bojana Peji c Here are some texts that cover Jordan canonical form: 1. Matrix Analysis, by Horn and Johnson, 2. Line
Math 2370 Matrices and Linear Operators Solutions and Hints to Practice Problems 2 Bojana Peji c Problem 1. Let k1 , . . . , kn be scalars such that k1 (x2 - x1 ) + k2 (x3 - x2 ) + + kn-1 (xn-1 - xn )