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 = cfw_ (a1 , a2) : a1 , a2 R . Define addition of elements
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 a1 v1 + . . . + an vn = 0 Since cfw_v1 , . . . , vn S
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 independent. The set in (c) is again independent. Problem 2
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), compute the nullity and rank of T, and verify the dimension
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 generates V . Suppose that c1 (u + v ) + c2 (au) = 0,
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 any , R.) Problem 4. 17 in Section 2.1. Problem 5. 18
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 transformation T : Rn Rm , compute [T] b. T : R2 R3 dene
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 be dim V there), f) False (T could take everything to ze
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: Using the associativity of matrix multiplication, (AB )(B -
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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 = cz1 . The matrix 1 [T ] is . 1
10. This is the matrix w
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 changes -coordinates into coordinates. b. = cfw_(-1, 3),
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 = I , so AB is invertible, with inverse equal to B 1 A1
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 invertible matrix Q and a diagonal matrix D such that Q-1 AQ =
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 by 0) e) True f) False (for instance, the system x + y
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 down from rst pivot: 1 ( 2x1 -2 ( x1 -1 ( 2x1 7 x2 2 x2 x
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 vectors in the resulting basis to obtain an orthonorma
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 vector y such that g(x) = x, y for all x V: Solution:
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 multiplicity m2 of 2 is at most 1 (the two multiplicities must
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 such that
1 N
1 1 1 < 2 . +1 n n
1 n
< . Then we have, |
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
A house allocation problem (Hylland & Zeckhauser, JPE 1979)
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 for V . (2) The set of all (3 3) matrices having trace
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 result we have proved in class or that is proved in the boo
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 initial value problem: y + 4y + 4y = 0, y (0) = y (0) =
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 proved in class or that is proved in the book. The numbers,
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