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 that if X is a finite-dimensional space then the space L(
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. I may not have time to type up the solutions to Probl
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(n - 1) 1 = n! different permutations. If (a1 a2 . . . a
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 T , then T is a scalar multiple of the
identity. We sho
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 way the reasoning behind important steps in the proofs
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 nonzero scalar valued function, there exists x0 X such
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 X into itself has a right (or left) inverse, then it i
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 Laplace expansion along the first column (several times)
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 students' homework assignments - good job! Problem 1. (
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, y) = 0, for all y Y , so x Y . Similarly x Z . Conver
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 use the fact that the minimal polynomial of T divides
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 for this question.) We have M = Pm = cfw_x P | deg(x) <
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 be defined such that x + y = (a1 + b1 , a 2 + b2 ) kx =
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 space K (i.e., set of sequences ( a1 , a2 ,.) of numbers
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 degree less than m, is P/M finite-dimensional ? How about if
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's rule to solve the following system over the field of
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 a scalar multiple of the identity map. Problem 2: Recal
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 finite-dimensional Euclidean space into itself then (a) tr (
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 space of Pn complex polynomials of degree less than n
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. Linear Algebra, by Hoffman and Kunze, 3. Linear Algebra Don