LINEAR ALGEBRA MATH 2700.006 SPRING 2013 (COHEN) LECTURE NOTES 1 Sets and Set Notation. Definition 1 (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most often name sets using capital letters, like A, B, X, Y , etc., while the elements of a set will usually be given lower-case letters, like x, y, z, v, etc. Two sets X and Y are called equal if X and Y consist of exactly the same elements. In this case we write X = Y . Example 1 (Examples of Sets). (1) Let X be the collection of all integers greater than or equal to 5 and strictly less than 10. Then X is a set, and we may write: X = f5; 6; 7; 8; 9g The above notation is an example of a set being described explicitly, i.e. just by listing out all of its elements. The set brackets {· · ·} indicate that we are talking about a set and not a number, sequence, or other mathematical object. (2) Let E be the set of all even natural numbers. We may write: E = f0; 2; 4; 6; 8; :::g This is an example of an explicity described set with infinitely many elements. The ellipsis (:::) in the above notation is used somewhat informally, but in this case its meaning, that we should \continue counting forever," is clear from the context. (3) Let Y be the collection of all real numbers greater than or equal to 5 and strictly less than 10. Recalling notation from previous math courses, we may write: Y = [5; 10) This is an example of using interval notation to describe a set. Note that the set Y obviously consists of infinitely many elements, but that there is no obvious way to write down the elements of Y explicitly like we did the set E in Example (2). Even though [5; 10) is a set, we don't need to use the set brackets in this case, as interval notation has a well-established meaning which we have used in many other math courses. (4) Now and for the remainder of the course, let the symbol ; denote the empty set, that is, the unique set which consists of no elements. Written explicitly, ; = f g. (5) Now and for the remainder of the course, let the symbol N denote the set of all natural numbers, i.e. N = f0; 1; 2; 3; :::g. (6) Now and for the remainder of the course, let the symbol R denote the set of all real numbers. We may think of R geometrically as being the collection of all the points on the number line. 1 2 LINEAR ALGEBRA MATH 2700.006 SPRING 2013 (COHEN) LECTURE NOTES (7) Let R2 denote the set of all ordered pairs of real numbers. That is, let R2 be the set which consists of all pairs (x; y) where x and y are both real numbers. We may think of R2 geometri- cally as the set of all points on the Cartesian coordinate plane. If (x; y) is an element of R2, it will often be convenient for us to write the pair as the column x vector . For our purposes the two notations will be interchangeable. It is important to y x y note here that the order matters when we talk about pairs, so in general we have 6= . y x (8) Let R3 be the set of all ordered triples of real numbers, i.e. R3 is the set of all triples (x; y; z) such that x, y, and z are all real numbers. R3 may be visualized geometrically as the set of all points in 3-dimensional Euclidean coordinate space. We will also write elements (x; y; z) of R3 2 x 3 be using the column vector notation 4 y 5. z (9) Lastly and most generally, let n ≥ 1 be any natural number. We will let Rn be the set of all ordered n-tuples of real numbers, i.e. the set of all n-tuples (x1; x2; :::; xn) for which each 2 3 x1 6 x2 7 coordinate xi, 1 ≤ i ≤ n, is a real number. We will also use the column vector notation 6 7 4 ::: 5 xn in this context. Definition 2 (Set Notation). If A is a set and x is an element of A, then we write: x 2 A. If B is a set such that every element of B is an element of A (i.e. if x 2 B then x 2 A), then we call B a subset of A and we write: B ⊆ A. In order to distinguish particular subsets we wish to talk about, we will frequently use set-builder notation, which for convenience we will describe informally using examples, rather than give a formal definition. For an example, suppose we wish to formally describe the set E of all even positive integers (See Example 1 (2)). Then we may write E = fx 2 N : x is evenly divisible by 2g. The above notation should be read as The set of all x in N such that x is evenly divisible by 2, which clearly and precisely defines our set E. For another example, we could write Y = fx 2 R : 5 ≤ x < 10g, which reads The set of all x in R such that 5 is less than or equal to x and x is strictly less than 10. The student should easily verify that Y = [5; 10) from Example 1 (3). In general, given a set A and a precise mathematical sentence P (x) about a variable x, the set-builder notation should be read as follows. f x 2 A : P (x)g \The set of all elements x in A such that sentence P (x) is true for the element x. LINEAR ALGEBRA MATH 2700.006 SPRING 2013 (COHEN) LECTURE NOTES 3 2 Vector Spaces and Subspaces. Definition 3. A (real) vector space is a nonempty set V , whose elements are called vectors, together with an operation +, called addition, and an operation ·, called scalar multiplication, which satisfy the following ten axioms: Addition Axioms. (1) If ~u 2 V and ~v 2 V , then ~u + ~v 2 V . (Closure under addition.) (2) ~u + ~v = ~v + ~u for all ~u;~v 2 V . (Commutative property of addition.) (3) (~u + ~v) + ~w = ~u + (~v + ~w) for all ~u;~v; ~w 2 V . (Associative property of addition.) (4) There exists a vector ~0 2 V which satisfies ~u + ~0 = ~u for all ~u 2 V . (Existence of an additive identity.) (5) For every ~u 2 V , there exists a vector −~u 2 V such that ~u + (−~u) = ~0. (Existence of additive inverses.) Scalar multiplication axioms. (6) If ~u 2 V and c 2 R, then c · ~u 2 V . (Closure under scalar multiplication.) (7) c · (~u + ~v) = c · ~u + c · ~v for all c 2 R, ~u;~v 2 V . (First distributive property of multiplication over addition.) (8) (c + d) · ~u = c · ~u + d · ~u for all c; d 2 R, ~u 2 V . (Second distributive property of multiplication over addition.) (9) c · (d · ~u) = (c · d) · ~u for all c; d 2 R, ~u 2 V . (Associative property of scalar multiplication.) (10) 1 · ~u = ~u for every ~u 2 V . We use the \arrow above" notation to help differentiate vectors (~u, ~v, etc.), which may or may not be real numbers, from scalars, which are always real numbers. When no confusion will arise, we will often drop the · symbol in scalar multiplcation and simply write c~u instead of c · ~u, c(~u + ~v) instead of c · (~u + ~v), etc. Example 2. Let V be an arbitrary vector space. (1) Prove that ~0 + ~u = ~u for every ~u 2 V . (2) Prove that the zero vector ~0 is unique. That is, prove that if ~w 2 V has the property that ~u + ~w = ~u for every ~u 2 V , then we must have ~w = ~0. (3) Prove that for every ~u 2 V , the additive inverse −~u is unique. That is, prove that if ~w 2 V has the property that ~u + ~w = ~0, then we must have ~w = −~u. Proof. (1) By Axiom (2), the commutativity of addition, we have ~0 + ~u = ~u + ~0. Hence by Axiom (4), we have ~0 + ~u = ~u + ~0 = ~u. 4 LINEAR ALGEBRA MATH 2700.006 SPRING 2013 (COHEN) LECTURE NOTES (2) Suppose ~w 2 V has the property that ~u + ~w = ~u for every ~u 2 V . Then in particular, we have ~0 + ~w = ~0. But ~0 + ~w = ~w by part (1) above; so ~w = ~0 + ~w = ~0. (3) Let ~u 2 V , and suppose ~w 2 V has the property that ~u + ~w = ~0. Let −~u be the additive inverse of ~u guaranteed by Axiom (5). Adding −~u to both sides of the equality above, and applying Axioms (2) and (3) (commutativity and associativity), we get −~u + (~u + ~w) = −~u + ~0 (−~u + ~u) + ~w = −~u (~u + (−~u)) + ~w = −~u ~0 + ~w = −~u: Now by part (1) above, we have ~w = ~0 + ~w = −~u. Example 3 (Examples of Vector Spaces). (1) The real number line R is a vector space, where both + and · are interpreted in the usual way. In this case the axioms are the familiar properties of real numbers which we learn in elementary school. x (2) Consider the plane 2 = : x; y 2 . Define an operation + on 2 by the rule R y R R x z x + z + = for all x; y; z; w 2 , y w y + w R and a scalar multiplication · by the rule x cx c · = for every c 2 , x; y 2 .