Ch. 1
Dept. of ECE, Faculty of Engineering University of Tehran
Linear Algebra Homework # I
Chapter 1: Vector Spaces Exercises to be handed: marked by *. Due date: Sunday Esfand 9th 1383. 1. Let V = c
Math 115A Homework 3 Solutions
Brett Hemenway April 28, 2006
1. Let S1 S2 V . (a) Suppose S1 is linearly dependent. This means we can nd v1 , . . . , vn S1 and a1 , . . . , an F not all zero such that
MATH 223 - HOMEWORK #3 Solutions Problem 1. 2 (a,b,c) in Section 1.5. The set in (a) is linearly dependent because the second matrix is 2 times the rst. In (b) there is no relation, so the set is inde
Homework 1 Solutions
Josh Hernandez October 27, 2009
1
2.1 - Linear Transformations, Null Spaces, and Ranges
For 3 and 6, prove that T is a linear transformation, nd bases for both N(T) and R(T), comp
MATH 115 SOLUTION SET 3
1.6:
10a, b. Im sure you can do these.
11. Suppose cfw_u, v is a basis for V . We claim that the set cfw_u + v, au is also a basis. We must show it is linearly independent and
MATH 223 - HOMEWORK #4 Due Friday, Oct 12 Problem 1. 2 in Section 2.1. Problem 2. 5 in Section 2.1. Problem 3. 10 in Section 2.1. (Note that if we know T (v) and T (w), then we also know T (v + w) for
Homework 4 Solutions
Josh Hernandez October 27, 2009
2.2 - The Matrix Representation of a Linear Transformation
2. Let and be the standard ordered bases for Rn and Rm , respectively. For each linear t
MATH 115 SOLUTION SET 4
ANSWERS TO SELECTED PROBLEMS
2.1:
1. a) True, b) False, c) False (look at T : R R dened by T (x) = x + 1; but also note that this is not linear.) d) True, e) False (it should b
Homework 5 Solutions
Josh Hernandez November 4, 2009
2.4 - Invertibility and Isomorphisms
4. Let A and B be n n invertible matrices. Prove that AB is invertible and (AB )-1 = B -1 A-1 . Solution: Usin
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MATH 115 SOLUTION SET 5
ANSWERS TO SELECTED PROBLEMS
2.2:
9. The fact that T is linear follows from standard facts about complex numbers: If z1 , z2 C then z1 + z2 = z1 + z2 . Also, if c R then cz1 =
Homework 6 Solutions
Joshua Hernandez November 11, 2009
2.5 - The Change of Coordinate Matrix
2. For each of the following pairs of ordered bases and for R2 , nd the change of coordinate matrix that c
MATH 115 SOLUTION SET 6
ANSWERS TO SELECTED PROBLEMS
2.4:
3. Only the pairs in b) and c) are isomorphic.
4. Since A and B are invertible, A1 and B 1 exist. We have (AB )(B 1 A1 ) = I and (B 1 A1 )AB =
Homework 7 Solutions
Joshua Hernandez November 16, 2009
5.2 - Diagonalizability
2. For each of the following matrices A Mnn (R), test A for diagonalizability, and if A is diagonalizable, nd an inverti
MATH 115 SOLUTION SET 2
1.4:
1. a) True b) False (the span is cfw_0) c) True (if you think super hard about this, youll see that it is the same as Theorem 1.5 in the book) d) False (you cant multiply
Homework 2 Solutions
Josh Hernandez October 27, 2009
1.4 - Linear Combinations and Systems of Linear Equations
2. Solve the following systems of linear equations. b. 2x1 x1 2x1 Solution: 1. Scaling do
Homework 8 Solutions
Joshua Hernandez November 18, 2009
6.2 - Gram-Schmidt Orthogonalization Process
2. Apply the Gram-Schmidt process to the given subset S of the inner product space V. Normalize the
Homework 9 Solutions
Joshua Hernandez December 6, 2009
6.3 - The Adjoint of a Linear Operator
2b. Let V = C2 (over C) and linear transformation g : V F dened by the mapping g(z1 , z2 ) = z1 2z2 , nd a
MATH 115 SOLUTION SET 8-10
ANSWERS TO SELECTED PROBLEMS
1. Set 8 5.2 8. We always have dim E2 1. On the other hand, since dim E1 = n 1, the algebraic multiplicity of 1 is at least n 1. So the multipli
Introduction to Analysis: Fall 2008 Practice problems V MTH 4101/5101 10/21/2008
1. Show that the sequence cfw_ (n21 converges to 0. +1) Solution: Let > 0 be given. For n I , we have N n2 Choose N su
Housing Markets & Top Trading Cycles Tayfun Snmez o
16th Jerusalem Summer School in Economic Theory Matching, Auctions, and Market Design
House Allocation Problems: A Collective Ownership Economy
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Practice Problems 2/13/06 (1) Let u and v be two distinct vectors of a vector space V . Let cfw_u, v be a basis for V and a, b nonzero scalars. Show that cfw_u + v, au and cfw_au, bv are also bases
MATH 223 - HOMEWORK #1 Due Friday, September 14 The problems in the book dene vector spaces over a eld F . You may assume that we always have F = R. When solving the problems you may refer to any resu
Math 135, HW 1
Due Wednesday, January 14th
1. Find the general solutions of the following equations: (a) y y = 0 (b) y + y = 0 (c) y + 4y + 4y = 0 (d) y + y = ex 2. Find the solution of the following
MATH 223 - HOMEWORK #1 Solutions The problems in the book dene vector spaces over a eld F . You may assume that we always have F = R. When solving the problems you may refer to any result we have prov
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