Review: Vector space

Size: px
Start display at page:

Download "Review: Vector space"

Transcription

1 Math 2F Linear Algebra Lecture 13 1 Basis and dimensions Slide 1 Review: Subspace of a vector space. (Sec. 4.1) Linear combinations, l.d., l.i. vectors. (Sec. 4.3) Dimension and Base of a vector space. (Sec. 4.4) Review: Vector space Slide 2 A vector space is a set of elements of any kind, called vectors, on which certain operations, called addition and multiplication by numbers, can be performed. The main idea in the definition of vector space is to do not specify the nature of the elements nor do we tell how the operations are to be performed on them. Instead, we require that the operations have certain properties, which we take as axioms of a vector space. Examples include spaces of arrows, matrices, functions.

2 Math 2F Linear Algebra Lecture 13 2 Review: Subspace Slide 3 Definition 1 (Subspace) A subspace W of a vector space V is a subset of V that is closed under the addition and scalar multiplication operations on V. That is, W V, and for all u, v W and a IR holds that u + v W, au W. Slide 4 Examples The set W IR 3 given by W = {x IR 3 : x = (x 1, x 2, ), is a subspace of IR 3. The set Ŵ IR3 given by Ŵ = {x IR 3 : x = (x 1, x 2, 1), in contrast is not a subspace of IR 3. x 1, x 2 IR}, x 1, x 2 IR},

3 Math 2F Linear Algebra Lecture 13 3 Examples The set W = {(x 1, x 2 ) IR 2 : x 1 } is not a subspace of IR 2. Slide 5 The segment W {x IR : 1 x 1} is not a subspace of IR. The line W = {x IR 3 : x = (1, 2, 3)t} is a subspace or IR 3. The line W = {x IR 3 : x = (1, 2, ) + (1, 2, 3)t} is not a subspace or IR 3. Linear combinations Definition 2 (Linear combination) Let V be a vector space, v 1,, v k V be arbitrary vectors, and a 1,, a k IR be arbitrary scalars. We call a linear combination of v 1,, v k the vector Slide 6 k a i v i = a 1 v a k v k. Linear combinations give rise to the concept of span of a set of vectors. Definition 3 (Span) The span of v 1,, v k V is the set of all linear combinations of these vectors, that is, { k } Span{v 1,, v k } = a i v i : a i IR, i k.

4 Math 2F Linear Algebra Lecture 13 4 Notice that every linear combination of v 1,, v k also belongs to V, so Span{v 1,, v k } V. One can also check that it is a subspace. Claim 1 Let V be a vector space and v 1,, v k V. Then, Span{v 1,, v k } V is a subspace. Slide 7 Different sets may span the same subspace. Example: The space IR 2 is spanned by {e 1, e 2 }, and also by {e 1, e 2, e 1 + 2e 2 }, and also by {, e 1 + 2e 2, e 1, e 1 + e 2 }. The space P 2 is spanned by {1, t, t 2 }, and also by {1, t, t 2, (t 2 1), (t + 1) 2 }. It is then useful to remove redundant vectors from linear combinations. Definition 4 (l.d.) Let V be a vector space. The vectors v 1,, v k V are linearly dependent, l.d., if there exist scalars a 1,, a k R with at least one of them nonzero, such that Slide 8 k a i v i =. Definition 5 (l.i.) Let V be a vector space. The vectors v 1,, v k V are linearly independent, l.i., if they are not l.d., that is, the only choice of scalars a 1,, a k R for which the equation k a i v i = holds is a i = for all 1 i k. Linearly independent set of vectors contain no redundant vectors, in the sense of linear combinations.

5 Math 2F Linear Algebra Lecture 13 5 Examples: If one element in S V is a scalar multiple of another, S is l.d.. If S, then S is l.d.. Slide 9 Let V be the space of continuous functions on [, 2π]. The set of vectors v 1 = cos 2 (t), v 2 = sin 2 (t), v 3 = 1 is l.d.. Indeed, Pythagoras theorem says that v 1 + v 2 v 3 =. Consider the space P 2. The set {1, t, t 2 } is l.i.. Indeed, the equation a + a 1 t + a 2 t 2 =, implies that all coefficient vanishes. Evaluate the equation at t =, then a =. So we have a 1 t + a 2 t 2 =. If the function vanishes, its derivative also vanishes. So a 1 + a 2 t =. Evaluate this at t =, then a 1 =. Repeat the procedure one more time, then a 2 =. Dimensions and base Definition 6 (Finite dimension and base) A vector space V has finite dimension is there exists a maximal set of l.i. vectors {v 1,, v n }. Slide 1 The number n is called the dimension of V, and we denote by n = dim V. The l.i. vectors {v 1,, v n } are called a base of V. If there is no maximal set of l.i. vectors, then V is called infinite dimensional. That is, the set {v 1,, v n } is maximal if it is l.i. but the set {v 1,, v n, v n+1 } is l.d. for all v n+1 V.

6 Math 2F Linear Algebra Lecture 13 6 Examples: The set {e 1, e 2 } IR 2 is a basis. Another basis is {e 1 + e 2, e 1 e 2 }. The set {e 1, e 2, e 1 + e 2 } is not a basis. The set {e 1, 2e 1 } is not a basis. Notice that a basis of a vector space is not unique. Slide 11 The space IR n is finite dimensional, of dimension n, because the vectors {e 1,, e n } are l.i., and any set of n + 1 vectors in IR n is l.d.. The space P, polynomials on [, 1] is infinite dimensional. The infinite set {1, t, t 2, t 3, } is l.i.. Theorem 1 A set of vectors E V is a base if the vectors in E are l.i. and span V. Components or coordinates Slide 12 Theorem 2 Let V be a finite dimensional vector space with n = dim V. If {u 1,, u n } is a basis of V, then each vector v V has a unique decomposition v = n a i u i. The n scalars a i are called components or coordinates of v with respect to this basis.

7 Math 2F Linear Algebra Lecture 14 7 Slide 13 Exercises Let u 1 = (1, 1) and u 2 = (1, 1), be a basis of IR 2. Find the components of x = (1, 2) in that basis. The vectors u 1, u 2 form a basis so there exists constants c 1, c 2 such that x = c 1 u 1 + c 2 u 2. That is, so c 1 = 3/2 and c 2 = 1/2, and then x = 3/2u 1 1/2u 2. c 1 c 2 = 1. 2 Then one has to solve the augmented matrix , Proof: The set {u 1,, u n } is a basis so there exists scalars a i, for 1 i n such that the following decomposition holds, v = n a i u i. This decomposition is unique. Because, if there is another decomposition v = n b i u i. then the difference has the form n (a i b i )u i =. Because the vectors {u 1,, u n } are l.i. this implies that a i = b i for all i n.

8 Math 2F Linear Algebra Lecture 14 8 Null and Column spaces Null space and range. (Sec. 4.2) Slide 14 Examples. Linear transformations. Null space and range for L.T.. Fundamental theorem of algebra. (Sec. 4.6) Null and range spaces: Matrix version Slide 15 Definition 7 Let A be an m n. The null space of A, denoted as N(A) V, is the set of all elements of V solution of Av =, that is, N(A) = {v V : Av = }. Definition 8 Let A be an m n. The column space of A, denoted as Col(A), is the span of the columns of A. Its dimension is called the rank of A, that is rank(a) = dim Col(A).

9 Math 2F Linear Algebra Lecture 14 9 Exercises Find the null space of A = Slide 16 The null space are all x IR 3 solutions of Ax =. One can check that these solutions have the form 3 e = 1 x 3. 1 Therefore, N(A) = Span{( 3, 1, 1)}. Exercises (Problem 2, Sec. 4.2) Find k 1 and k 2 such that N(A) is a subspace of IR k1, and Col(A) is a subspace of IR k2, with A = [1, 3, 9,, 5]. The solutions x of the equation Ax = form a hyperplane in IR 5, given by the equation Slide 17 x 1 3x 2 + 9x 3 5x 5 =. So, the solutions can be written as x = x x 3 + x x 5. 1 N(A) = Span{(3, 1,,, ), ( 9,, 1,, ), (,,, 1, ), (5,,,, 1)} IR 4. Col(A) IR.

10 Math 2F Linear Algebra Lecture 14 1 linear transformations Definition 9 Let V, W be vector spaces. The function T : V W is called a linear transformation if it has the following properties: T (v 1 + v 2 ) = T (v 1 ) + T (v 2 ) for all v 1, v 2 V ; T (av) = at (v), for all v V and a IR. Slide 18 These properties means that T preserves addition and multiplication by numbers. The two properties combine in one formula T (av + bu) = at (v) + bt (u), or equivalently, ( k ) k T a i v i = a i T (v i ). Examples: The identity transformation, that is T : V V, given by T (v) = v. Slide 19 A stretching by a IR, that is, T : V V, given by T (v) = av. A projection, that is T : IR 3 IR 3 given by T (x 1 e 1 + x 2 e 2 + x 3 e 3 ) = x 1 e 1 + x 2 e 2. The derivative. Let V be the vector space of differentiable functions on (, 1), and let W be the vector space of continuous function on (, 1). Then, T : V W given by T (f) = f is a linear transformation.

11 Math 2F Linear Algebra Lecture Null space and range of T Slide 2 Definition 1 (Null space of T ) Let T : V W be a linear transformation. The null space of T, denoted as N(T ) V, is the set of all elements of V that T maps onto the W. That is, N(T ) = {v V : T (v) = }. Definition 11 (Range of T ) Let T : V W be a linear transformation. The range of T, denoted as T (V ) W, is the set of all elements of W of the form w = T (v) for some v V. Theorem 3 (N(T ) subspace of V ) Let T : V W be a linear transformation. The null space of T is a subspace of V. Slide 21 Theorem 4 (T (V ) subspace of W ) Let T : V W be a linear transformation. The range of T, denoted by T (V ) W, is a subspace of W. The nullity is the dimension of N(T ). The rank is the dimension of T (V ). Theorem 5 Let T be the linear transformation associated to A. Then N(T ) = N(A) and T (V ) = Col(A).

12 Math 2F Linear Algebra Lecture Proof of Theorem 3: Assume that u and v N(T ). Then, au + bv N(T ) because T (au + bv) = at (u) + bt (v) = a + b =. Proof of Theorem 4: Given two vectors T (u) and T (v) in the range of T, then at (u) + bt (v) also belongs to the range of T because T is linear, that is, at (u) + bt (v) = T (au + bv). Examples: The identity transformation. The null space is N(I) = {}. The range space is T (V ) = V. Slide 22 A stretching by a IR. In the case a the null space is N(T ) = {} and T (V ) = V. In the case a = one has N(T ) = V and T (V ) = {}. A projection, T : IR 3 IR 3 given by T (x 1 e 1 + x 2 e 2 + x 3 e 3 ) = x 1 e 1 + x 2 e 2. Then N(T ) = {x 3 e 3 }. The range space is the plane spanned by {e 1, e 2 }. The derivative, given by T (f) = f. Then N(T ) are the constant functions. The range of T are the continuous functions on (, 1).

13 Math 2F Linear Algebra Lecture Fundamental theorem of algebra Slide 23 Theorem 6 (Nullity plus rank) Let T : V W be a linear transformation, and V be finite dimensional. Then N(T ) and T (V ) are finite dimensional and the following relation holds, dim N(T ) + dim T (V ) = dim V. Theorem 7 (Nullity plus rank: Matrix form) Let A be an m n matrix with m, n finite Then, the following relation holds, dim N(A) + rank(a) = n. Proof of Theorem 6: Let n = dim V and let S = {e 1,, e k } be a basis for N(T ), so we say that the nullity is some number k. Because N(T ) is contained in V one knows that k n. Let us add l.i. vectors to S to complete a basis of V, say, {e 1,, e k, e k+1,, e k+r } for some number r such that k + r = n. We shall prove that {T (e k+1 ),, T (e k+r )} is a basis for T (V ), and then r = dim T (V ). This relation proves Theorem 6. The elements {T (e k+1 ),, T (e k+r )} are a basis of T (V ) because they span T (V ) and they are l.i.. They span T (V ) because for every w T (V ) we know that there exists v V such that w = T (v). If we write v = n v ie i, then we have ( n ) n k+r w = T e i = a i T (e i ) = a i T (e i ), i= i= i=k+1 then the {T (e k+1 ),, T (e k+r )} span T (V ). These vectors are also l.i., by the following argument. Suppose there are scalars c k+1,, c k+r such that Then, this implies T k+r i=k+1 ( k+r i=k+1 c i T (e i ) =. c i e i ) =, so the vector u = k+r i=k+1 c it (e i ) belongs to N(T ). But if u belongs to N(T ), then it must be written also as a linear combination of the elements of the base of N(T ), namely, the

14 Math 2F Linear Algebra Lecture vectors e 1,, e k, so there exists constants c 1,, c k such that u = k c i e i. Then, we can construct the linear combination = u u = k c i e i k+r i=k+1 Because the set {e 1, e k+r } is a basis o V we have that all the c i with 1 i k + r must vanish. Then, the vectors {T (e k+1 ),, T (e k+r )} are l.i.. Therefore they are a basis of T (V ), and then the dimension of dim T (V ) = r. This proves the Theorem. c i e i. The matrix of a linear transformation Slide 24 Theorem 8 Let T : V W be a linear transformation, where V and W are finite dimensional vector spaces with dim V = n and dim W = m. Let {v 1,, v n } be a basis of V, and {w 1,, w m } be a basis of W. Then, T has associated a matrix A = [a i ], 1 i n, given by m T (v i ) = a i = a ij w j. j=1 Idea of the proof: every vector in W is a linear combinations of the basis vectors w j for j m. In particular all the n vectors T (e i ), i n. therefore, there must exist the m n set of numbers a ij given above. These coefficients define the matrix A associated to T.

15 Math 2F Linear Algebra Lecture Example: In the case T : IR 2 IR 3, one chooses the basis vectors v 1 = (1, ), v 2 = (, 1), for IR 2, and the basis vectors w 1 = (1,, ), w 2 = (, 1, ), w 3 = (,, 1) for IR 3. Then, a linear transformation Slide 25 T (x 1, x 2 ) = (x 1 + 3x 2, x 1 + x 2, x 2 ), has associated the matrix A = [a 1, a 2 ] given by 1 3 a 1 = 1, a 2 = 1. 1 Invertible transformations Slide 26 One-to-one and onto. Isomorphisms (one-to-one and onto). Invertible transformations.

16 Math 2F Linear Algebra Lecture One-to-one and onto Definition 12 (one-to-one onto) Let T : V W be a linear transformation. Slide 27 T is one-to-one if v 1 v 2 implies T (v 1 ) T (v 2 ); T is onto if for all w W there exists v V such that T (v) = w. Theorem 9 Let T : V W be a linear transformation. T is one-to-one N(T ) = {}; T is onto T (V ) = W. Isomorphism Definition 13 (Isomorphism) The linear transformation T : V W is an isomorphism if T is one-to-one and onto. Isomorphic spaces are essentially the same. There is a one to one correspondence between elements from one space and of the other. Slide 28 Example: P 2 is isomorphic to R 3. A basis of P 2 is {1, t, t 2 } A general element v P has the form v = a + a 1 t + a 2 t 2. The linear transformation T : P 2 R 3 given by T (a + a 1 t + a 2 t 2 ) = a e + a 1 e 2 + a 2 e 3 IR 3 is an isomorphism. Theorem 1 Let T : V W be an isomorphism, with V and W of finite dimension. Then dim V = dim W.

NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

More information

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column

More information

Lecture Note on Linear Algebra 15. Dimension and Rank

Lecture Note on Linear Algebra 15. Dimension and Rank Lecture Note on Linear Algebra 15. Dimension and Rank Wei-Shi Zheng, wszheng@ieee.org, 211 November 1, 211 1 What Do You Learn from This Note We still observe the unit vectors we have introduced in Chapter

More information

Sec 4.1 Vector Spaces and Subspaces

Sec 4.1 Vector Spaces and Subspaces Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common

More information

Orthogonal Diagonalization of Symmetric Matrices

Orthogonal Diagonalization of Symmetric Matrices MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding

More information

Methods for Finding Bases

Methods for Finding Bases Methods for Finding Bases Bases for the subspaces of a matrix Row-reduction methods can be used to find bases. Let us now look at an example illustrating how to obtain bases for the row space, null space,

More information

4.1 VECTOR SPACES AND SUBSPACES

4.1 VECTOR SPACES AND SUBSPACES 4.1 VECTOR SPACES AND SUBSPACES What is a vector space? (pg 229) A vector space is a nonempty set, V, of vectors together with two operations; addition and scalar multiplication which satisfies the following

More information

T ( a i x i ) = a i T (x i ).

T ( a i x i ) = a i T (x i ). Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)

More information

Inner product. Definition of inner product

Inner product. Definition of inner product Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product

More information

Recall that two vectors in are perpendicular or orthogonal provided that their dot

Recall that two vectors in are perpendicular or orthogonal provided that their dot Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

More information

4.5 Linear Dependence and Linear Independence

4.5 Linear Dependence and Linear Independence 4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then

More information

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

More information

Similarity and Diagonalization. Similar Matrices

Similarity and Diagonalization. Similar Matrices MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

More information

These axioms must hold for all vectors ū, v, and w in V and all scalars c and d.

These axioms must hold for all vectors ū, v, and w in V and all scalars c and d. DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms

More information

MA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam

MA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam MA 4 LINEAR ALGEBRA C, Solutions to Second Midterm Exam Prof. Nikola Popovic, November 9, 6, 9:3am - :5am Problem (5 points). Let the matrix A be given by 5 6 5 4 5 (a) Find the inverse A of A, if it exists.

More information

α = u v. In other words, Orthogonal Projection

α = u v. In other words, Orthogonal Projection Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v

More information

MATH1231 Algebra, 2015 Chapter 7: Linear maps

MATH1231 Algebra, 2015 Chapter 7: Linear maps MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter

More information

( ) which must be a vector

( ) which must be a vector MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

More information

Systems of Linear Equations

Systems of Linear Equations Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

More information

4.6 Null Space, Column Space, Row Space

4.6 Null Space, Column Space, Row Space NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear

More information

Chapter 20. Vector Spaces and Bases

Chapter 20. Vector Spaces and Bases Chapter 20. Vector Spaces and Bases In this course, we have proceeded step-by-step through low-dimensional Linear Algebra. We have looked at lines, planes, hyperplanes, and have seen that there is no limit

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

More information

1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0

1 0 5 3 3 A = 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 Solutions: Assignment 4.. Find the redundant column vectors of the given matrix A by inspection. Then find a basis of the image of A and a basis of the kernel of A. 5 A The second and third columns are

More information

Math 312 Homework 1 Solutions

Math 312 Homework 1 Solutions Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

More information

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued). MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0

More information

MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =

MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A = MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the

More information

Solutions to Math 51 First Exam January 29, 2015

Solutions to Math 51 First Exam January 29, 2015 Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not

More information

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1. MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

More information

1 Sets and Set Notation.

1 Sets and Set Notation. LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

More information

Matrix Representations of Linear Transformations and Changes of Coordinates

Matrix Representations of Linear Transformations and Changes of Coordinates Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under

More information

Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)

Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of

More information

Lecture Notes 2: Matrices as Systems of Linear Equations

Lecture Notes 2: Matrices as Systems of Linear Equations 2: Matrices as Systems of Linear Equations 33A Linear Algebra, Puck Rombach Last updated: April 13, 2016 Systems of Linear Equations Systems of linear equations can represent many things You have probably

More information

Inner Product Spaces and Orthogonality

Inner Product Spaces and Orthogonality Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,

More information

by the matrix A results in a vector which is a reflection of the given

by the matrix A results in a vector which is a reflection of the given Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

More information

Name: Section Registered In:

Name: Section Registered In: Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are

More information

2.1: MATRIX OPERATIONS

2.1: MATRIX OPERATIONS .: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and

More information

Linear Algebra Notes

Linear Algebra Notes Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note

More information

Vector Spaces II: Finite Dimensional Linear Algebra 1

Vector Spaces II: Finite Dimensional Linear Algebra 1 John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition

More information

Vector Spaces 4.4 Spanning and Independence

Vector Spaces 4.4 Spanning and Independence Vector Spaces 4.4 and Independence October 18 Goals Discuss two important basic concepts: Define linear combination of vectors. Define Span(S) of a set S of vectors. Define linear Independence of a set

More information

Linear Dependence Tests

Linear Dependence Tests Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

More information

MATH 110 Spring 2015 Homework 6 Solutions

MATH 110 Spring 2015 Homework 6 Solutions MATH 110 Spring 2015 Homework 6 Solutions Section 2.6 2.6.4 Let α denote the standard basis for V = R 3. Let α = {e 1, e 2, e 3 } denote the dual basis of α for V. We would first like to show that β =

More information

1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES 1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

More information

Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.

Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively. Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry

More information

Inner Product Spaces

Inner Product Spaces Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

More information

1 Introduction to Matrices

1 Introduction to Matrices 1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns

More information

MATH 551 - APPLIED MATRIX THEORY

MATH 551 - APPLIED MATRIX THEORY MATH 55 - APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points

More information

1 Eigenvalues and Eigenvectors

1 Eigenvalues and Eigenvectors Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x

More information

THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

More information

Math 333 - Practice Exam 2 with Some Solutions

Math 333 - Practice Exam 2 with Some Solutions Math 333 - Practice Exam 2 with Some Solutions (Note that the exam will NOT be this long) Definitions (0 points) Let T : V W be a transformation Let A be a square matrix (a) Define T is linear (b) Define

More information

MATH10212 Linear Algebra B Homework 7

MATH10212 Linear Algebra B Homework 7 MATH22 Linear Algebra B Homework 7 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra and its Applications Pearson, 26 (or other editions) Normally, homework assignments

More information

Math 240: Linear Systems and Rank of a Matrix

Math 240: Linear Systems and Rank of a Matrix Math 240: Linear Systems and Rank of a Matrix Ryan Blair University of Pennsylvania Thursday January 20, 2011 Ryan Blair (U Penn) Math 240: Linear Systems and Rank of a Matrix Thursday January 20, 2011

More information

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal

More information

160 CHAPTER 4. VECTOR SPACES

160 CHAPTER 4. VECTOR SPACES 160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results

More information

MATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3

MATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3 MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................

More information

NOTES on LINEAR ALGEBRA 1

NOTES on LINEAR ALGEBRA 1 School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura

More information

GROUP ALGEBRAS. ANDREI YAFAEV

GROUP ALGEBRAS. ANDREI YAFAEV GROUP ALGEBRAS. ANDREI YAFAEV We will associate a certain algebra to a finite group and prove that it is semisimple. Then we will apply Wedderburn s theory to its study. Definition 0.1. Let G be a finite

More information

Solution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A.

Solution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A. Solutions to Math 30 Take-home prelim Question. Find the area of the quadrilateral OABC on the figure below, coordinates given in brackets. [See pp. 60 63 of the book.] y C(, 4) B(, ) A(5, ) O x Area(OABC)

More information

2.1 Functions. 2.1 J.A.Beachy 1. from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair

2.1 Functions. 2.1 J.A.Beachy 1. from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 2.1 J.A.Beachy 1 2.1 Functions from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 21. The Vertical Line Test from calculus says that a curve in the xy-plane

More information

ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES OF R n KEITH CONRAD ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x

More information

INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL

INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics

More information

LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA. September 23, 2010 LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

More information

Definition of a Linear Program

Definition of a Linear Program Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

More information

Linearly Independent Sets and Linearly Dependent Sets

Linearly Independent Sets and Linearly Dependent Sets These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation

More information

Chapter 19. General Matrices. An n m matrix is an array. a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm. The matrix A has n row vectors

Chapter 19. General Matrices. An n m matrix is an array. a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm. The matrix A has n row vectors Chapter 9. General Matrices An n m matrix is an array a a a m a a a m... = [a ij]. a n a n a nm The matrix A has n row vectors and m column vectors row i (A) = [a i, a i,..., a im ] R m a j a j a nj col

More information

Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232

Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232 Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232 In mathematics, there are many different fields of study, including calculus, geometry, algebra and others. Mathematics has been

More information

5. Orthogonal matrices

5. Orthogonal matrices L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 5-1 Orthonormal

More information

The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14

The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14 4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with

More information

LS.6 Solution Matrices

LS.6 Solution Matrices LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions

More information

1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each)

1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each) Math 33 AH : Solution to the Final Exam Honors Linear Algebra and Applications 1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each) (1) If A is an invertible

More information

Section 6.1 - Inner Products and Norms

Section 6.1 - Inner Products and Norms Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,

More information

Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

More information

is in plane V. However, it may be more convenient to introduce a plane coordinate system in V.

is in plane V. However, it may be more convenient to introduce a plane coordinate system in V. .4 COORDINATES EXAMPLE Let V be the plane in R with equation x +2x 2 +x 0, a two-dimensional subspace of R. We can describe a vector in this plane by its spatial (D)coordinates; for example, vector x 5

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

More information

MAT188H1S Lec0101 Burbulla

MAT188H1S Lec0101 Burbulla Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u

More information

Linear Algebra Review. Vectors

Linear Algebra Review. Vectors Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length

More information

Orthogonal Projections

Orthogonal Projections Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors

More information

Applied Linear Algebra I Review page 1

Applied Linear Algebra I Review page 1 Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties

More information

MAT 242 Test 2 SOLUTIONS, FORM T

MAT 242 Test 2 SOLUTIONS, FORM T MAT 242 Test 2 SOLUTIONS, FORM T 5 3 5 3 3 3 3. Let v =, v 5 2 =, v 3 =, and v 5 4 =. 3 3 7 3 a. [ points] The set { v, v 2, v 3, v 4 } is linearly dependent. Find a nontrivial linear combination of these

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

More information

MA106 Linear Algebra lecture notes

MA106 Linear Algebra lecture notes MA106 Linear Algebra lecture notes Lecturers: Martin Bright and Daan Krammer Warwick, January 2011 Contents 1 Number systems and fields 3 1.1 Axioms for number systems......................... 3 2 Vector

More information

Math 215 HW #6 Solutions

Math 215 HW #6 Solutions Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T

More information

Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions

Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential

More information

Solutions to Linear Algebra Practice Problems

Solutions to Linear Algebra Practice Problems Solutions to Linear Algebra Practice Problems. Find all solutions to the following systems of linear equations. (a) x x + x 5 x x x + x + x 5 (b) x + x + x x + x + x x + x + 8x Answer: (a) We create the

More information

Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 313 Lecture #10 2.2: The Inverse of a Matrix Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

More information

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function 17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function, : V V R, which is symmetric, that is u, v = v, u. bilinear, that is linear (in both factors):

More information

Linear Algebra. A vector space (over R) is an ordered quadruple. such that V is a set; 0 V ; and the following eight axioms hold:

Linear Algebra. A vector space (over R) is an ordered quadruple. such that V is a set; 0 V ; and the following eight axioms hold: Linear Algebra A vector space (over R) is an ordered quadruple (V, 0, α, µ) such that V is a set; 0 V ; and the following eight axioms hold: α : V V V and µ : R V V ; (i) α(α(u, v), w) = α(u, α(v, w)),

More information

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

More information

Lecture 6. Inverse of Matrix

Lecture 6. Inverse of Matrix Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that

More information

On the representability of the bi-uniform matroid

On the representability of the bi-uniform matroid On the representability of the bi-uniform matroid Simeon Ball, Carles Padró, Zsuzsa Weiner and Chaoping Xing August 3, 2012 Abstract Every bi-uniform matroid is representable over all sufficiently large

More information

Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

More information

Lecture 14: Section 3.3

Lecture 14: Section 3.3 Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in

More information

LEARNING OBJECTIVES FOR THIS CHAPTER

LEARNING OBJECTIVES FOR THIS CHAPTER CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional

More information

PROJECTIVE GEOMETRY. b3 course 2003. Nigel Hitchin

PROJECTIVE GEOMETRY. b3 course 2003. Nigel Hitchin PROJECTIVE GEOMETRY b3 course 2003 Nigel Hitchin hitchin@maths.ox.ac.uk 1 1 Introduction This is a course on projective geometry. Probably your idea of geometry in the past has been based on triangles

More information

some algebra prelim solutions

some algebra prelim solutions some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no

More information

3. INNER PRODUCT SPACES

3. INNER PRODUCT SPACES . INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

More information

Section 2.1. Section 2.2. Exercise 6: We have to compute the product AB in two ways, where , B =. 2 1 3 5 A =

Section 2.1. Section 2.2. Exercise 6: We have to compute the product AB in two ways, where , B =. 2 1 3 5 A = Section 2.1 Exercise 6: We have to compute the product AB in two ways, where 4 2 A = 3 0 1 3, B =. 2 1 3 5 Solution 1. Let b 1 = (1, 2) and b 2 = (3, 1) be the columns of B. Then Ab 1 = (0, 3, 13) and

More information

13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.

13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions. 3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms

More information

Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.

Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve

More information