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### Homework I

Course: MATH math115a, Spring 2010
School: UCLA
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1 Dept. Ch. 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 = { (a1 , a2) : a1 , a2 R }. Define addition of elements of V coordinate-wise, and for (a1 , a2) in V and c R, define: (0 , 0) c(a1 , a 2 ) = (ca1 , a 2 c) if c = 0 if c 0 . Is V a vector space under these...

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1 Dept. Ch. 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 = { (a1 , a2) : a1 , a2 R }. Define addition of elements of V coordinate-wise, and for (a1 , a2) in V and c R, define: (0 , 0) c(a1 , a 2 ) = (ca1 , a 2 c) if c = 0 if c 0 . Is V a vector space under these operations? Justify your answer. 2. Let V be the set of sequences {an} of real numbers. For any {an} and {bn} V and any real number , define {an} + {bn} = {an + bn} and .{an} = {.an}. Prove that V is a vector space under these operations. 3. Let V and W be vector spaces over a field F. Let Z = { (v , w) : v V and w W }. Prove that Z is a vector space over F under the operations (v1 , w1) + (v2 , w2) = (v1 + v2 , w1 + w2) and c.(v , w) = (c.v , c.w) . 4. How many elements are there in the vector space Mmxn(Z2) ? 5. Prove that the upper triangular matrices form a subspace of Mmxn(F). Notice: An mxn matrix A is called upper triangular if aij = 0 for i > j. 6. Let W1 and W2 be subspaces of a vector space V. Prove that W1 U W2 is a subspace of V if and only if W1 W2 or W2 W1. *7. Show that the set of convergent sequences {an} is a subspace of the vector space V in exercise 2. Ch. 1 8. Let W1 denote the set of all polynomials f(x) in P(F) such that in the representation f(x) = an xn + an-1 xn-1 + + a0, we have aj = 0 whenever j is even. Likewise let W2 denote the set of all polynomials g(x) in P(F) such that in the representation g(x) = bn xn + bn-1 xn-1 + + b0, we have bj = 0 whenever j is odd. Prove that P(F) = W1 W2. 9. In Mmxn(F) define W1 denote the set of upper triangular matrices and W2 = { A Mmxn(F) : aij = 0 whenever i j}. Show that Mmxn(F) = W1 W2. *10. Let W be a subspace of a vector space V over a field F. For any v V let set {v} + W = { v + w : w W } is called the coset of W containing v. It is customary to denote v + W rather than {v} + W. Prove the following: a) v + W is a subspace of V if and only if v W. b) v1 + W = v2 + W if and only if v1 v2 W. *11. Let S denotes the collection (or class) of all cosets of W a subspace of vector space V over filed F. Addition and multiplication by elements of field F can be defined in S as follows: (v1 + W) + (v2 + W) = (v1 + v2) + W a.(v + W) = a.v + W v1 and v2 V, v V and a F. a) Prove that the operations above are well-defined; i.e., show that if v11 + W = v12 + W and v21 + W = v22 + W, then and (v11 + W) + (v21 + W) = (v12 + W) + (v22 + W) a.(v11 + W) = a.(v12 + W) a F. b) Prove that S is a vector space under the operations defined above. This vector space is called the quotient space of V modulo W is and denoted by V W . 12. Show that if S1 and S2 are arbitrary subsets of a vector space V, then: L (S1 U S2) = L (S1) + L (S2). Ch. 1 13. Let S1 and S2 be subsets of a vector space V. Prove that L (S1 I S2) L (S1) I L (S2). *14. Let S = { u1 , u2 , u3 , ., un } be a linearly independent subset of a vector space V over field Z2. How many elements are there in L (S)? Justify your answer. 15. Let V be a vector space over field F, and let u and v be distinct vectors in V. Prove that { u , v } is linearly independent if and only if { v + u and v u } are linearly independent. *16. Let M be a square upper triangular matrix with non-zero diagonal entries. Prove that columns of M are linearly independent. 17. Let S be a set of non-zero polynomials in P(F) such that no two elements of S has the same degree. Prove that S is linearly independent. 18. Prove that if { A1 , A2, . , Ak } is a linearly independent subset of Mnxn(F), then { A1T , A2T, . , AkT } is also linearly independent. 19. Let u and v be distinct elements of a vector space V. Show that if { u , v } is a basis for V and a and b are non-zero scalars, then both { u + v , a.u } and { a.u , b.v } are also bases for V. *20. Fins bases for the following subspaces of F5: W1 = { (a1 , a2 , a3 , a4 , a5) F5 : a1 a3 a4 = 0} and W2 = { (a1 , a2 , a3 , a4 , a5) F5 : a2 = a3 = a4 and a1 + a5 = 0 }. What are the dimensions of W1 and W2? 21. Prove that the set of all nxn matrices having trace equal to zero (denoted by W) is a subspace of Mnxn(F). Find a basis for W. What is the dimension of W? Ch. 1 *22. Let V be a vector space having dimension n, and let S be a subset of V that generates V. a) Prove that there is a subset of S that is a basis for V (be careful not to assume that S is finite). b) Prove that S contains at least n elements. *23. Let f(x) be a polynomial of degree n in Pn(R). Prove that for any g(x) Pn(R) there exist scalars c0 , c1 , , cn such that g(x) = c0f(x) + c1f(1)(x) + c2f(2)(x) + . + cnf(n)(x), where f(j)(x) denotes the jth derivatives of f(x). *24. Prove that if W1 is any subspace of a finite-dimensional vector space V, then there exists a subspace W2 of V such that V = W1 W2. *25. Let W be a subspace of a finite-dimensional vector space V, and consider the basis { u1 , u2 , , uk } for W. Let { u1 , u2 , , uk , uk+1 , uk+2 , , un } be an extension of this basis to a basis for V. Prove that { uk+1 + W , uk+2 + W, , un + W} is a basis for V W . 26. Let B be a subset of an infinite-dimensional vector space V. Then B is a basis for V if and only if for each non-zero vector v in V, there exist unique vectors u1 , u2 , , un in B and non-zero scalars c1 , c2 , , cn such that v = c1u1 + c2u1 + + cnun .
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UCLA - MATH - math115a
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
UCLA - MATH - math115a
Math 135, HW 1Due Wednesday, January 14th1. 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) =
UCLA - MATH - math115a
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,
UCLA - MATH - math115a
UCLA - MATH - math115a
Homework 1 SolutionsJosh Hernandez October 25, 20091.1 - Introduction2. Find the equations of the lines through the following pairs of points in space. b. (3, -2, 4) and (-5, 7, 1) Solution: x = (3, -2, 4) + r[(-5, 7, 1) (3, -2, 4)] = (3, -2, 4) + r(-8
UCLA - MATH - math115a
MATH 115 SOLUTION SET 11.2: 1, 4. See back of book.7. To show that f = g , it is enough to show that f (x) = g (x) for all x S . Since f (0) = g (0) = 1 and f (1) = g (1) = 3. we have f = g . Similarly for f + g = h.8. By the rst distributive law, (a +
UCLA - MATH - math115a
Math 135, HW 2Due Wednesday, January 21st1. Show that L[cos ax] = by integrating. 2. Show that L[cosh ax] = without integrating. 3. Find L[sin2 ax] without integrating. 4. Find a function whose Laplace transform is: (a) (b)12 p5 1 p3 +pp2p ,p &gt; 0 + a
UCLA - MATH - math115a
MATH 223 - HOMEWORK #2 Due Friday, September 21 Problem 1. 8(a,b,f) in Section 1.3. Problem 2. 11 in in Section 1.3. Problem 3. Let P7 be the vector space of polynomials of degree 7 or less. Dene W = cfw_p(x) P7 |p(1) = p(2) = . . . = p(5) = 0. (1)Show th
UCLA - MATH - math115a
MATH 223 - HOMEWORK #2 Solutions Problem 1. 8(a,b,f) in Section 1.3. A subspace is defined by linear equations with right hand side equal to zero. The equations in (a) are of this type if we bring all a-s to one side. The equations in (b) and (f) do not d
UCLA - MATH - math115a
UCLA - MATH - math115a
Homework 2 SolutionsJosh Hernandez October 27, 20091.4 - Linear Combinations and Systems of Linear Equations2. 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
UCLA - MATH - math115a
UCLA - MATH - math115a
MATH 115 SOLUTION SET 21.4:1. 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
UCLA - MATH - math115a
Math 115A Homework 3 SolutionsBrett Hemenway April 28, 20061. 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
UCLA - MATH - math115a
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
UCLA - MATH - math115a
UCLA - MATH - math115a
Homework 1 SolutionsJosh Hernandez October 27, 200912.1 - Linear Transformations, Null Spaces, and RangesFor 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
UCLA - MATH - math115a
MATH 115 SOLUTION SET 31.6:10a, 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,
UCLA - MATH - math115a
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
UCLA - MATH - math115a
UCLA - MATH - math115a
Homework 4 SolutionsJosh Hernandez October 27, 20092.2 - The Matrix Representation of a Linear Transformation2. 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
UCLA - MATH - math115a
MATH 115 SOLUTION SET 4ANSWERS TO SELECTED PROBLEMS2.1:1. 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
UCLA - MATH - math115a
Homework 5 SolutionsJosh Hernandez November 4, 20092.4 - Invertibility and Isomorphisms4. 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 -
UCLA - MATH - math115a
UCLA - MATH - math115a
MATH 115 SOLUTION SET 5ANSWERS TO SELECTED PROBLEMS2.2:9. 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 . 110. This is the matrix w
UCLA - MATH - math115a
Homework 6 SolutionsJoshua Hernandez November 11, 20092.5 - The Change of Coordinate Matrix2. 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),
UCLA - MATH - math115a
UCLA - MATH - math115a
MATH 115 SOLUTION SET 6ANSWERS TO SELECTED PROBLEMS2.4:3. 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
UCLA - MATH - math115a
Homework 7 SolutionsJoshua Hernandez November 16, 20095.2 - Diagonalizability2. 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 =
UCLA - MATH - math115a
MATH 115 SOLUTION SET 7ANSWERS TO SELECTED PROBLEMS5.17. The point of this problem is to extend the denition of determinant from matrices to linear transformations. The result is always that a property satised by matrices is always satised by linear tr
UCLA - MATH - math115a
Homework 8 SolutionsJoshua Hernandez November 18, 20096.2 - Gram-Schmidt Orthogonalization Process2. 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
UCLA - MATH - math115a
Homework 9 SolutionsJoshua Hernandez December 6, 20096.3 - The Adjoint of a Linear Operator2b. 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:
UCLA - MATH - math115a
MATH 115 SOLUTION SET 8-10ANSWERS TO SELECTED PROBLEMS1. 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
UCLA - MATH - math115a
Introduction to Analysis: Fall 2008 Practice problems V MTH 4101/5101 10/21/20081. Show that the sequence cfw_ (n21 converges to 0. +1) Solution: Let &gt; 0 be given. For n I , we have N n2 Choose N such that1 N1 1 1 &lt; 2 . +1 n n1 n&lt; . Then we have, |
UCLA - MATH - math115a
Housing Markets &amp; Top Trading Cycles Tayfun Snmez o16th Jerusalem Summer School in Economic Theory Matching, Auctions, and Market DesignHouse Allocation Problems: A Collective Ownership EconomyA house allocation problem (Hylland &amp; Zeckhauser, JPE 1979)
UCLA - MATH - math115a
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
Stony Brook University - CHE - CHE 133
Final Exercise 105 pointsGravimetric Determination of NaHCO3 Click to edit Master subtitle style in a MixtureLast Update: 2/2/10 07:09:28 AMCHE 133 MAKE-UP LABORATORY EXERCISE Friday (tomorrow) After Quiz, before checl out either a TEST (105 point) or
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Gasometric Determination of NaHCO3 in a MixtureClick to edit Master subtitle styleLast Update: 2/2/10 07:10:04 AMPurpose: Determine the percent of NaHCO3 in a mixture by Gasometry Concepts: Ideal Gas LawHenrys Law Vapor Pressure Stoichiometry Technique
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Strength of Vinegar by Acid-Base TitrationFinal Exercise 105 points Click to edit Master subtitle styleLast Update: 2/2/10 07:10:05 AM? QUESTIONS ?With what accuracy and precision can you determine the concentration of a solution of sodium hydroxide?
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I was wondering if mutation could not be a potential cause of evolution by itself. This is Question #30 from the exam starting with &quot;Consider a gene&quot;: For Question #30, my thinking was mutation only provided the raw material for evolution. Without natural
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