Introduction to Vector Spaces and Subspaces Sections 4.1-4.2, 4.7 Math 81, Applied Analysis Instructor: Dr. Doreen De Leon 1 Why Study Vector Spaces? Vector spaces have many applications. For one, they provide a framework to deal with analytical and geometrical problems, and are used in the Fourier transform. There are also applications of vector spaces in optimization theory. The minimax theorem of game theory stating the existence of a unique payoff when all players play optimally can be formulated and proven using vector space methods. Least squares estimation, which is used in among other areas digital filter design, tracking (Kalman filters), control systems, etc. Representation theory basically transforms problems in abstract algebra, particularly group theory, to problems in linear algebra. Such discussions are, however, beyond the scope of this class. Therefore, we limit ourselves to the above brief discussion, and launch directly into the topic at hand. 2 Introduction to Vector Spaces Sections 4.1-4.2, 4.7 2.1 Vectors in R n We will use some familiar ideas about vectors to motivate the idea of a vector space, which we will introduce a bit later. First, a vector can be thought of as a directed line segment (or arrow) that has both magnitude and direction. In Calculus III, you probably defined vector addition in two ways: using the parallelogram law, where x + y is the diagonal of the parellelogram formed by x and y; and by adding corresponding elements of x and y to obtain x + y. Then, to form the sum of three vectors, we simply add two vectors as above and then add the third vector to the result. This can be performed equivalently by adding the first two vectors together, and then adding the third vector to the sum; or, by adding the last two 1
vectors together, and then adding the first vector to the sum. So, we have that for all vectors x, y, and z x + y = y + x, x + (y + z) = (x + y) + z. The zero vector, denoted 0, is defined as the vector satisfying x + 0 = x, for all vectors x. We consider the zero vector as having zero magnitude and arbitrary direction. Geometrically, we picture the zero vector as corresponding to a point in space. Let x denote the vector that has the same magnitude as x but points in the opposite direction. Then, if we add x and x using either approach above, we obtain x + ( x) = 0. The vector x so defined is the additive inverse of x. fundamental properties of vector addition. The above properties are the We also need to define the operation of multiplication of a vector by a scalar, i.e., scalar multiplication. Geometrically, if x is a vector and c is a scalar, then cx is defined by the vector whose magnitude is c times the magnitude of x and whose direction is the same as x if c > 0 and the opposite of x if c < 0. If c = 0, then cx = 0. Scalar multiplication has several important properties, which can easily be verified, either geometrically or algebraically using the definitions of vector addition and scalar multiplication. For all vectors x and y and all scalars a, and b, 1x = x, (ab)x = a(bx), a(x + y) = ax + ay, (a + b)x = ax + bx. If you notice, we did not define multiplication of vectors. However, from Calculus III, you are familiar with the dot product and cross product of vectors. For the purpose of talking about vector spaces, we will ignore these two operations on vectors, only concerning ourselves with vector addition and scalar multiplication. If we consider vectors in the plane, denoted R 2, then each vector v = (v 1, v 2 ) can be identified with a point in space with coordinates (v 1, v 2 ), and the direction and magnitude of v are determined by considering the arrow with vertex at the origin, (0, 0). Then, for all vectors v = (v 1, v 2 ) and w = (w 1, w 2 ) and all scalars a, we define vector addition and scalar multiplication as v + w = (v 1, v 2 ) + (w 1, w 2 ) = (v 1 + w 1, v 2 + w 2 ), av = a (v 1, v 2 ) = (av 1, av 2 ). We can easily verify the above listed properties for vectors in R 2. For example, 0 = (0, 0) and since (v 1, v 2 ) + ( v 1, v 2 ) = (0, 0), 2
the additive inverse of v is v = ( v 1, v 2 ). It is straightforward to extend these ideas to vectors in space, denoted R 3 by simply adding a third component. Each vector v = (v 1, v 2, v 3 ) can be identified with a point in space with coordinates (v 1, v 2, v 3 ), and the direction and magnitude of v are determined by considering the arrow with vertex at the origin, (0, 0, 0). Vector addition and scalar multiplication are defined as follows. If v = (v 1, v 2, v 3 ) and w = (w 1, w 2, w 3 ) and c is a scalar, v + w = (v 1, v 2, v 3 ) + (w 1, w 2, w 3 ) = (v 1 + w 1, v 2 + w 2, v 3 + w 3 ), av = c (v 1, v 2, v 3 ) = (cv 1, cv 2, cv 3 ). And again, we can show that the properties given above for vector addition and scalar multiplication are satisfied by R 3, with 0 = (0, 0, 0) and v = ( v 1, v 2, v 3 ). We can generalize this notion of vectors that we can visualize to vectors with any number of components. The set of all vectors with n real-valued components is denoted by R n. Vector addition and scalar multiplication in R n are defined componentwise. Given any two vectors u and v in R n, defined by u = (u 1, u 2,..., u n ) and v = (v 1, v 2,..., v n ), and any scalar c, the standard vector addition and scalar multiplication are defined as follows: u + v = (u 1 + v 1, u 2 + v 2,..., u n + v n ) cu = (cu 1, cu 2,..., cu n ) We can show that for any positive integer n, R n possesses all of the algebraic properties for vector addition and scalar multiplication as R 2 and R 3. These properties give us the framework to define a general notion, that of vector spaces. 2.2 Vector Spaces We will start with a set of elements V, which we will call vectors. They may be vectors in the sense of vectors discussed previously, or they may be some other mathematical construct, such as functions, matrices, or polynomials. Associated with this set of vectors are two operations, vector addition and scalar multiplication. Vector addition can be thought of as a rule for combining two vectors in V. To simplify things notationally, we will use the + sign to denote vector addition. So, the result of adding two vectors u and v will be denoted u + v. Scalar multiplication can be thought of as a rule for combining each vector in V with any scalar, and we will use the usual notation av to denote the result of multiplying the vector v by the scalar a. So, what is a vector space? 3
Definition: Let V be a set of elements called vectors, in which the operations of addition of vectors and multiplication of vectors by scalars are defined. Then, given any vectors u, v, w V and any scalars a and b, V is a vector space if all of the following properties are satisfied. (1) u + v V (V is closed under vector addition). (2) au V (V is closed under scalar multiplication). (3) u + v = v + u (commutativity). (4) u + (v + w) = (u + v) + w (associativity). (5) There exists a zero element in V, 0 V, such that u + 0 = 0 + u = u. (6) There exists an element in V, u, called the additive inverse, such that u + ( u) = u + u = 0. (7) a(u + v) = au + av. (8) (a + b)u = au + bu. (9) a(bu) = (ab)u. (10) (1)u = u. Notation: 4
2.3 Subspaces Definition: Theorem 1. Examples: 1) W = {(x, y, z) : z = 0}. Is W a subspace of R 3? Why or why not? 5
2) W = {(x, y, z) : y = 1} Is W a subspace of R 3? Why or why not? 3) W = {(x, y, z) : x 2 + y 2 z 2 = 0}. Is W a subspace of R 3? Why or why not? 6
2.4 Solution Subspaces Theorem 2. If A is a constant m n matrix, then the solution set of the system is a subspace of R n, called the solution space of the system. Ax = 0 (1) Proof. Let W be the set of all solutions to (1). Verify the conditions for a subspace. 2.5 General Vector Spaces The term vector in vector space can be interpreted in a more general sense. Examples: (1) Given m and n positive integers, define M mn as the set of all m n matrices with real entries. Then M mn is a vector space. Matrices play the role of vectors, with matrix addition defining vector addition multiplication of a matrix by a scalar defining scalar multiplication The zero element is the zero matrix. It is easily verified using these definitions of vector addition and scalar multiplication that M mn satisfies properties (1)-(10) of the definition of a vector space. 7
(2) F is the set of all real-valued functions defined on R. Functions play the role of vectors, with vector addition and scalar multiplication defined as follows. For f, g F, (f + g)(x) = f(x) + g(x) vector addition For f F and c R, (cf)(x) = cf(x) scalar multiplication. It is easily verified using these definitions of vector addition and scalar multiplication that F is a vector space, with the zero element being the function f(x) = 0 (i.e., the function whose value is zero for all x). (3) C n [a, b] is the set of all continuous functions with n continuous derivatives on [a, b]. C n [a, b] is a vector space with the same properties as F. (4) P is the set of all polynomials. P is a vector space with the polynomials playing the role of vectors. Note: P is also a subspace of F since P F. (5) P n is the set of all polynomials of degree at most n. P n is a vector space with the polynomials playing the role of vectors. In fact, P n is a subspace of P. 2.6 Subspaces of General Vector Spaces 8
Examples: (1) W is the set of diagonal 2 2 matrices. Is W a subspace of M 22? Why or why not? (2) W is the set of all solutions of the differential equation y + p(x)y = 0. Is W a subspace of C 1 (I)? Why or why not? 9
(3) W is the set of all solutions of the differential equation y + p(x)y = x. Is W a subspace of C 1 (I)? Why or why not? (4) W is the set of all polynomials of the form a 0 + a 1 x + a 2 x 2 such that a 0 = 2a 2. Is W a subspace of P 2? Why or why not? 10
(5) W is the set of all polynomials in P 2 whose coefficients are odd integers. Is W a subspace of P 2? Why or why not? 11