SYMBOL EXPLANATION EXAMPLE

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1 MATH 4310 PRELIM I REVIEW Notation These are the symbols we have used in class, leading up to Prelim I, and which I will use on the exam SYMBOL EXPLANATION EXAMPLE {a, b, c, } The is the way to write the set containing the elements a, b, c, etc {1, 2} is the set whose elements are the numbers 1 and 2 or or \ N, N Z, Z Read this symbol as is an element of a A means that a is an element of the set A a, b A means that both a and b are elements of the set A In the above example, 1, 2 {1, 2} These are symbols for subset You can use If A = {1, 5} and B = {1, 5, 16}, then A B either If A and B are sets then A B if all of the elements of A are also elements of B This is the symbol for union If A and B are sets then A B is the set that consists of all of the elements in A and all of the elements in B This is the symbol for intersection If A and B are sets then A B is the set that consists of all of the elements that are in both of A and B These are symbols for set minus If A and B are sets then B A (or B \ A) is the set which consists of the elements of B which are NOT elements of A If A = {1, 2} and B = {2, 3, 4}, then A B = {1, 2, 3, 4} If A = {1, 2} and B = {2, 3, 4}, then A B = {2} If A = {1, 2} and B = {2, 3, 4}, then B A = {3, 4} This is the set of natural numbers, ie N = {0, 1, 2, } 17, 24, , N This is the set of integers, ie Z = {, 2, 1, 0, 1, 2, } 63, 24, Z Q, Q This is the set of rational numbers, ie fractions of integers If a, b Z and b 0, then a b Q We have 1 2, 2 17 Q Note that Z Q 1

2 SYMBOL EXPLANATION EXAMPLE R, R C, C F, F V S(a 1,, a n ) S + T A = (α i,j ) v w This is the set of real numbers, numbers that may be written in decimal form We did not give a more rigorous definition of the real numbers We have assumed the basic properties of arithmetic in the real number system This is the set of complex numbers, numbers that may be written in the form a + b 1 where a, b R The number a is the real part, and the number b is the imaginary part We sometimes denote i = 1 We may add complex numbers by adding the real and imaginary components individually This denotes a field or number system that has two operations + and that satisfy seven axioms If you want to refer to the axioms, you may either refer to an axiom by the number we used in class, or by a description such as commutativity or existence of multiplicative inverses We denote general field elements by Greek letters α F This denotes a vector space (over a field F) together with two operations, + : V V V and : F V V, satisfying seven axioms We denote vectors by Roman letters v V This represents the subspace of all linear combinations λi a i of a 1,, a n This denotes the sum of two subspaces S and T of a vector space V It is again a subspace of V This denotes a matrix with entries in a field F The set of m n matrices with entries in F is denoted M m n (F) This denotes the dot product of two vectors v, w F n, n defined by (α 1,, α n ) (β 1,, β n ) = α i β i i=1 3, 1, 1 2, 2, π R Note that Q R 3, 1, 1 2, 2, π, 2 3 i, 1 C Note that R C Examples include Q, R, C, and the integers modulo a prime F p You may use on the exam that these are fields Neither N nor Z is a field! F n, Pol n (F), Pol(F) = F[x], F un(x, F) are all vector spaces over the field F We have a few special examples for the field F = R: C 0 (R), C k (R), C (R) Let e 1,, e n denote the standard basis vectors in F n That is, e i is the vector with a single non-zero entry in the i th coordinate Then F n = S(e 1,, e n ) The xy-plane in R 3 is the sum of S = {(α, 0, 0} and T = {(0, β, 0)} [ 1 1 π 0 ] [ M 2 2 (R) and (1, 1, 0) (1, 1, 0) = = 2 in R 3, and (1, 1, 0) (1, 1, 0) = = 0 in (F 2 ) 3 ] M 2 3 (F 7 ) 2

3 Material we have covered This is a brief summary of the material we have covered, and that you are responsible for knowing for Prelim I The material we have covered corresponds to 1 9 in our textbook Linear Algebra, An Introductory Approach, by Charles W Curtis It is listed here in a slightly different order from how we covered it in class The axioms we assume are the ground rules we agree upon For us, we have agreed on the properties of arithmetic of the real and complex numbers We are also assuming some familiar properties of sets, as laid out in the notation tables above Methods of proof We specifically discussed direct proofs and proofs by induction A fields is a set with two operations, + and, satisfying by seven axioms We deduced a number of basic properties from the axioms A vector space over a field F is a set V of vectors, together with two operations, vector addition + : V V V and scalar multiplication : F V V, satisfying seven axioms We deduced a number of basic properties of vector spaces from the axioms A subspace of a vector space is a subset of vectors that is closed under vector addition and scalar multiplication We defined linear combinations of a set of vectors {v 1,, v n }, and showed that the set of all linear combinations of {v 1,, v n } is a subspace, denoted S(v 1,, v n ) It is the smallest subspace of V that contains the set {v 1,, v n } We defined linearly dependent and linearly independent sets of vectors For a single vector, it is linearly independent if and only if it is not zero Two vectors are linearly independent if and only if they are not scalar multiples of one another A subspace S V is generated by or spanned by a set of vectors {a 1,, a n } if it is the space of all linear combinations of {a 1,, a n }, S = S(a 1,, a n ) If S is spanned by n vectors, then for any m > n, any set of m vectors in S will necessarily be linearly dependent A set of vectors {b 1,, b k } V is called a basis for V if it is a linearly independent set and it spans V We proved that every basis has the same number of vectors, and we define the dimension of a vector space to be * 0 if V = {0} (and we declare S( ) = {0}, so is a basis); * k if there is a basis of V containing k elements; and * infinite, for all other vector spaces Every finitely generated vector space has a basis, and the basis can be chosen to be a subset of any generating set Thus, every spanning set containing exactly dim(v ) vectors must be a basis In a finite dimensional vector space, any list of linearly independent vectors may be extended to be a basis The vectors you must add can be chosen from a prescribed basis Thus, every linearly independent set containing exactly dim(v ) vectors must be a basis The sum S + T of two vector subspaces S and T of V is again a subspace of V, and its dimension satisfies dim(s + T ) = dim(s) + dim(t ) dim(s T ) A matrix is an m n array of scalars α i,j F It has m rows r 1,, r m that can be thought of as vectors in F n, and n columns c 1,, c n that can be thought of as vectors in F m The row space of A is R(A) = S(r 1,, r m ) F n, and the column space is C (A) = S(c 1,, c n ) F m We recalled some properties and algorithms that you should have seen in Math 2210/2940 (at least over R or C), and discussed how they work when the entries are in a more general field A matrix is in row echelon form if * all non-zero rows of the matrix are at the top of the matrix; * the left-most non-zero entry in a row (called the leading coefficient or pivot) is strictly to the right of the pivot in the row above; and * all entries below a pivot are zero A matrix is in reduced row echelon form if, in addition, * the pivots are all 1, and the pivot column has a single non-zero entry (the pivot) 3

4 We may use elementary row operations to transform a matrix into one that is in (reduced) row echelon form They are I Exchange two rows; II Add µ times one row to another row, for µ F; and III Multiply a row by a non-zero scalar 0 µ F Two matrices are row equivalent if there is a sequence of elementary row operations starting from one and ending at the other Row equivalent matrices have the same row space The algorithm to transform a matrix into one in (reduced) row echelon form systematically is called row reduction or Gaussian elimination A system of m linear equations in n unknowns is a system of equations α 1,1 x α 1,n x n = β 1 α m,1 x α m,n x n = β m for scalars α i,j, β k F We may represent this by putting the scalars α i,j into a matrix A M m n (F), and the variables and βs into vectors x = x 1 x n and b = β 1 β n, and writing Ax = b Alternatively, if c 1,, c n are the columns of A, this is equivalent to the vector equation x 1 c x n c n = b Finally, if the rows of A are r 1,, r m, then the system of equations may be written using the dot product as r 1 x = β 1 r m x = β m The dot product is an operation : F n F n F defined by (α 1,, α n ) (β 1,, β n ) = n α i β i A solution to the system is a vector l = (λ 1,, λ n ) F n that satisfies λ 1 c λ n c n = b; or equivalently satisfies r i l = β i for i = 1,, m We denote the set of all solutions to Ax = b by Sol(Ax = b) The rank of a matrix is the dimension of the column space C (A) We denote the rank by rank(a) Suppose that l is one solution to Ax = b Then the set of all solutions to Ax = b is the set Sol(Ax = b) = {l + m m Sol(Ax = 0)} The set Sol(Ax = 0) is a subspace of F n called the nullspace and denoted N(A) or Null(A) It has dimension dim(sol(ax = 0)) = n rank(a) The row rank of a matrix is the dimension of the row space R(A) We proved that the row rank of a matrix is equal to the rank i=1 4

5 Some sample true/false questions The first (in-class) prelim will consist of 5 6 true/false questions (where you do NOT need to give an explanation), followed by some questions that are similar to the problems in the homework You will have a choice in which questions you answer, you will probably have to do 3 of 5 problems Some sample true/false questions (HINT: five are true and seven are false) Every vector space contains a zero vector TRUE FALSE The empty set is a subspace of every vector space TRUE FALSE The intersection of any two subsets of V is a subspace of V TRUE FALSE If S is a linearly dependent set, then each element of S is a linear combination of other elements of S TRUE FALSE Any set containing the zero vector is linearly dependent TRUE FALSE Every vector space that is generated by a finite set has a basis TRUE FALSE A vector space cannot have more than one basis TRUE FALSE If a vector space has a finite basis, then the number of vectors in every basis is the same TRUE FALSE The dimension of Pol n (F) is n TRUE FALSE The dimension of M m n (F) is m + n TRUE FALSE If S = S(b 1,, b k ), then every vector in S can be written as a linear combination of b 1,, b k in only one way TRUE FALSE If V is a vector space of dimension n, then V has exactly one subspace of dimension 0 and exactly one subspace of dimension n TRUE FALSE 5