Math 550 Notes Chapter 1 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 1 Fall 2010 1 / 27 Outline 1 Complex Numbers 2 Definition of Vector Space 3 Properties of Vector Spaces 4 Subspaces 5 Sums and Direct Sums (Tarleton State University) Math 550 Chapter 1 Fall 2010 2 / 27 Binary Operations Definition A binary operation on a set S is a mapping from S × S to S. (Tarleton State University) Math 550 Chapter 1 Fall 2010 3 / 27 Fields Definition Suppose F is a set. Let + be a binary operation on F called addition. Let · be a binary operation of F called multiplication. Then F is called a field if + and · are commutative: x + y = y + x and xy = yx for all x; y 2 F: + and · are associative: (x + y) + z = x + (y + z) and (xy)z = x(yz) for all x; y; z 2 F: (Tarleton State University) Math 550 Chapter 1 Fall 2010 4 / 27 F contains an additive identity, 0, and a multiplicative identity, 1, and 0 6= 1: x + 0 = x and x · 1 = x for all x 2 F: Every x 2 F has an additive inverse −x 2 F: x + (−x) = 0: Every x 2 F n f0g has a multiplicative inverse x−1: x(x−1) = 1: The distributive law holds: x(y + z) = xy + xz for all x; y; z 2 F: (Tarleton State University) Math 550 Chapter 1 Fall 2010 5 / 27 Ordered Field Definition Suppose F is a field. Then F is called an ordered field if it contains a subset F +, such that F + is closed under addition and multiplication: If x; y 2 F + then x + y 2 F + and xy 2 F +; and for each x 2 F, exactly one of the following holds: x = 0; x 2 F +; or − x 2 F +: (Tarleton State University) Math 550 Chapter 1 Fall 2010 6 / 27 > and Upper Bounds Definition Suppose x and y are elements of an ordered field F. Then x > y if x − y 2 F +. Definition Let S be a subset of an ordered field F. u 2 F is called an upper bound of S if u ≥ x for all x 2 S: u 2 F is called a least upper bound of S if I u is an upper bound of S, and I for any upper bound u~ of S, u ≤ u~. (Tarleton State University) Math 550 Chapter 1 Fall 2010 7 / 27 Complete Ordered Fields Definition Let F be an ordered field. Then F is complete if every nonempty subset of F that has an upper bound in F has a least upper bound in F. Theorem The field of real numbers R is a complete ordered field. (Tarleton State University) Math 550 Chapter 1 Fall 2010 8 / 27 Field of Complex Numbers Definition The field of complex numbers C and its operations of + and · are defined as follows: C = f(a; b) j a; b 2 Rg (a; b) + (c; d) = (a + c; b + d) (a; b) · (c; d) = (ac − bd; ad + bc) Notation For any a 2 R, a ≡ (a; 0). i := (0; 1) (a; b) = a + bi i2 = −1 Theorem C is a field. (Tarleton State University) Math 550 Chapter 1 Fall 2010 9 / 27 Notation Throughout the book and notes, F stands for R or C. Elements of F are called scalars. − For x; y 2 F, x − y := x + (−y), and x=y := x · y 1. (Tarleton State University) Math 550 Chapter 1 Fall 2010 10 / 27 Outline 1 Complex Numbers 2 Definition of Vector Space 3 Properties of Vector Spaces 4 Subspaces 5 Sums and Direct Sums (Tarleton State University) Math 550 Chapter 1 Fall 2010 11 / 27 n-tuples Definition An n-tuple or list of length n is an ordered collection of n objects, (x1;:::; xn): Repetitions are allowed and order matters. Finite length The list can be empty. So, a 0-tuple is the empty list (). If we write x = (x1;:::; xn), then xj is the jth coordinate or component of x. Two n-tuples x and y are equal if xj = yj for all j = 1;:::; n: (Tarleton State University) Math 550 Chapter 1 Fall 2010 12 / 27 Fn Definition n F := f(x1;:::; xn) j xj 2 F for j = 1;:::; ng. Elements of Fn are called vectors. Addition is defined on Fn by (x1;:::; xn) + (y1;:::; yn) = (x1 + y1;:::; xn + yn) 0 ≡ (0;:::; 0) x + 0 = x for all x 2 Fn. 0 is an additive identity for Fn. Scalar multiplication is defined by n a · (x1;:::; xn) = (ax1;:::; axn), for all a 2 F and x 2 F : (Tarleton State University) Math 550 Chapter 1 Fall 2010 13 / 27 Vector Spaces Definition Suppose F is a field and V is a set. Let + be a binary operation on V called addition. Let · be a function from F × V to V called scalar multiplication. Then V is called a vector space over F if There exists an additive identity 0 2 V . Every x 2 V has an additive inverse −x 2 V . + is commutative and associative. 1 · x = x for all x 2 V . (ab)x = a(bx) for all a; b 2 F and x 2 V . (a + b)x = ax + bx and a(x + y) = ax + ay for all a; b 2 F and x; y 2 V . (Tarleton State University) Math 550 Chapter 1 Fall 2010 14 / 27 Definition Elements of V are called vectors or points. A vector space over R is called a real vector space, and a vector space over C is called a complex vector space. Fn is a vector space over F. (Tarleton State University) Math 550 Chapter 1 Fall 2010 15 / 27 More Examples of Vector Spaces Example 1 F := f(x1; x2;:::) j xj 2 F, for j = 1; 2;:::g (x1; x2;:::) + (y1; y2;:::) = (x1 + y1; x2 + y2;:::) a(x1; x2;:::) = (ax1; ax2;:::) Definition A function p : F ! F is called a polynomial if 2 m p(z) = a0 + a1z + a2z + ··· + amz ; for some a0;:::; am 2 F. Let P(F) denote the set of all polynomials with coefficients in F. P(F) is a vector space over F. (Under which operations?) (Tarleton State University) Math 550 Chapter 1 Fall 2010 16 / 27 Outline 1 Complex Numbers 2 Definition of Vector Space 3 Properties of Vector Spaces 4 Subspaces 5 Sums and Direct Sums (Tarleton State University) Math 550 Chapter 1 Fall 2010 17 / 27 Properties of Vector Spaces Notation For the rest of the book/notes, V denotes a vector space over F. Because + and · are associative, parentheses are not necessary in expressions such as u + v + w and abv; where u; v; w 2 V and a; b 2 F. Proposition V has a unique additive identity. Every element of V has a unique additive inverse. 0v = 0, for all v 2 V . a0 = 0, for all a 2 F. (−1)v = −v, for all v 2 V . (Tarleton State University) Math 550 Chapter 1 Fall 2010 18 / 27 Outline 1 Complex Numbers 2 Definition of Vector Space 3 Properties of Vector Spaces 4 Subspaces 5 Sums and Direct Sums (Tarleton State University) Math 550 Chapter 1 Fall 2010 19 / 27 Subspaces Definition Suppose U ⊆ V . Then U is a subspace of V if U is a vector space (under the same + and · operations that make V a vector space). Proposition (Subspace Criterion) If U ⊆ V , then U is a subspace of V iff 0 2 U, for all u; v 2 U, u + v 2 U, and for all a 2 F and u 2 U, au 2 U. (Tarleton State University) Math 550 Chapter 1 Fall 2010 20 / 27 Examples of Subspaces Example 1 fp 2 P(F) j p(3) = 0g is a subspace of P(F). 2 The subspaces of R2 are precisely f0g, all lines in R2 through the origin, and R2. 3 The subspaces of R3 are precisely f0g, all lines in R3 through the origin, all planes in R3 through the origin, and R3. 4 V and f0g are always subspaces of V . (Tarleton State University) Math 550 Chapter 1 Fall 2010 21 / 27 Outline 1 Complex Numbers 2 Definition of Vector Space 3 Properties of Vector Spaces 4 Subspaces 5 Sums and Direct Sums (Tarleton State University) Math 550 Chapter 1 Fall 2010 22 / 27 Sums Definition Suppose U1;:::; Um are subspaces of V . Then their sum is defined by U1 + ··· + Um = fu1 + ··· + um j u1 2 U1;:::; um 2 Umg: U1 + ··· + Um is the smallest subspace of V containing U1;:::; Um. Example Suppose U = f(x; 0; 0) 2 F3 j x 2 Fg, and W = f(0; y; 0) 2 F3 j y 2 Fg. Then U + W = f(x; y; 0) 2 F3 j x; y 2 Fg. Suppose W~ = f(y; y; 0) 2 F3 j y 2 Fg. Then U + W~ = f(x; y; 0) 2 F3 j x; y 2 Fg also. (Tarleton State University) Math 550 Chapter 1 Fall 2010 23 / 27 Direct Sums Definition Suppose U1;:::; Um are subspaces of V , and V = U1 + ··· + Um. Then V is the direct sum of U1;:::; Um if every element of V can be written uniquely in the form u1 + ··· + um; where uj 2 Uj , j = 1;:::; m. This is denoted by V = U1 ⊕ · · · ⊕ Um Example Both of the examples from the previous slide are direct sums. (Tarleton State University) Math 550 Chapter 1 Fall 2010 24 / 27 Example Define I U1 = f(x; y; 0) j x; y 2 Fg, I U2 = f(0; 0; z) j z 2 Fg, and I U3 = f(0; y; y) j y 2 Fg.