Math 28S Vector Spaces Fall 2011 Definition: Given a field F , a vector space over F is a set V together with two operations: • addition: + : V × V ! V (i.e. (u; v) 7! u + v) • scalar multiplication, F × V ! V (i.e. (c; v) 7! cv) such that the following rules (called the \Vector Space Laws") are satisfied: 1. Addition is closed: For all u; v 2 V , u + v 2 V . 2. Scalar multiplication is closed: For all c 2 F and all v 2 V , cv 2 V . 3. Addition is commutative: For all u; v 2 V , u + v = v + u. 4. Addition is associative: For all u; v; w 2 V , u + (v + w) = (u + v) + w. 5. Additive identity element: There exists an element of V called 0 such that v + 0 = v for all v 2 V . 6. Additive inverses exist: For all v 2 V , there exists an element −v 2 V such that v+(−v) = 0. 7. Distributivity I: For all c 2 F and all u; v 2 V , c(u + v) = cu + cv. 8. Distributivity II: For all c; d 2 F and all v 2 V ,(c + d)v = cv + dv. 9. Scalar multiplication is associative: For all c; d 2 F and all v 2 V ,(cd)v = c(dv). 10. Identity element for scalar multiplication: 1v = v where 1 refers to the multiplicative identity element of F . Language: Given a vector space V , the field F over which it lies is called the underlying field of V ; elements of the underlying field are called scalars; elements of the vector space are called vectors. Notation: Vectors are usually referred to by boldface letters (like v) when typed, and as let- ters with arrows over them (like −!v ) when hand-written. However, sometimes we get lazy and just −! refer to a vector with a letter (like v). The zero scalar is denoted 0; the zero vector is denoted 0 or 0 . Theorem: R2 = f(x; y): x; y 2 Rg is a vector space over R, where the addition and scalar multiplication are defined coordinate-wise, i.e. (x1; y1) + (x2; y2) = (x1 + x2; y1 + y2) and c(x1; y1) = (cx1; cy1): Proof: Let u; v; and w be arbitrary elements of R2, and let c; d 2 R. By definition of R2, we have u = (u1; u2); v = (v1; v2) and w = (w1; w2). We check the vector space laws one by one: 1. Addition is closed: This is obvious by the definition of vector addition. 2. Scalar multiplication is closed: This is obvious by the definition of scalar multiplication. 3. Addition is commutative: We need to check u + v = v + u: u + v = (u1; u2) + (v1; v2) 2 = (u1 + v1; u2 + v2) (by defn of + in R ) = (v1 + u1; v2 + u2) (since + is commutative in R) 2 = (v1; v2) + (u1; u2) (by defn of + in R ) = v + u: 4. Addition is associative: u + (v + w) = (u1; u2) + ((v1; v2) + (w1; w2)) 2 = (u1; u2) + (v1 + w1; v2 + w2) (by defn of + in R ) 2 = (u1 + (v1 + w1); u2 + (v2 + w2)) (by defn of + in R ) = ((u1 + v1) + w1; (u2 + v2) + w2) (since + is associative in R) 2 = (u1 + v1; u2 + v2) + (w1; w2) (by defn of + in R ) 2 = ((u1; u2) + (v1; v2)) + (w1; w2) (by defn of + in R ) = (u + v) + w: Math 28S Vector Spaces Fall 2011 5. Additive identity element: Let 0 = (0; 0). Then v + 0 = (v1; v2) + (0; 0) 2 = (v1 + 0; v2 + 0) (by defn of + in R ) = (v1; v2) (since 0 is additive identity in R) = v: 6. Additive inverses exist: Let −v = (−v1; −v2). Then v + (−v) = (v1; v2) + (−v1; −v2) 2 = (v1 + (−v1); v2 + (−v2)) (by defn of + in R ) = (0; 0) (by properties of additive inverses in R) = 0 (by defn of 0): 7. Distributivity I: c(u + v) = c((u1; u2) + (v1; v2)) 2 = c(u1 + v1; u2 + v2) (by defn of + in R ) = (c(u1 + v1); c(u2 + v2)) (by defn of scalar multiplication) = (cu1 + cv1; cu2 + cv2) (by distributivity of R) 2 = (cu1; cu2) + (cv1; cv2) (by defn of + in R ) = c(u1; u2) + c(v1; v2) (by defn of scalar multiplication) = cu + cv: 8. Distributivity II: For all c; d 2 F and all v 2 V ,(c + d)v = cv + dv. (c + d)v) = (c + d)(v1; v2) = ((c + d)v1; (c + d)v2) (by defn of scalar multiplication) = (cv1 + dv1; cv2 + dv2) (by distributivity of R) 2 = (cv1; cv2) + (dv1; dv2) (by defn of + in R ) = c(v1; v2) + d(v1; v2) (by defn of scalar multiplication) = cv + dv: 9. Scalar multiplication is associative: For all c; d 2 F and all v 2 V ,(cd)v = c(dv). (cd)v = (cd)(v1; v2) = ((cd)v1; (cd)v2) (by defn of scalar multiplication) = (c(dv1); c(dv2)) (by associativity of · in R) = c(dv1; dv2) (by defn of scalar multiplication) = c(d(v1; v2)) (by defn of scalar multiplication) = c(dv): 10. Identity element for scalar multiplication: 1v = 1(v1; v2) = (1v1; 1v2) (by defn of scalar multiplication) = (v1; v2) (since 1 is mult. identity in R) = v: 2 Since all the laws hold, R is indeed a vector space over R. Math 28S Typical Vector Spaces Fall 2011 n 1. \Traditional" vectors: Given any field F and any n 2 N, F = f(x1; :::; xn): xj 2 F 8 jg is a vector space over F , where the addition and scalar multiplication are defined coordinate-wise. (The proof of this is the same as the proof that R2 is a vector space, essentially.) 2. Fields: Given any field F , F is a vector space over itself (where the addition and scalar multiplication are the field operations). In particular, F = F 1. 3. Zero vector spaces: Given any field F , f0g is a vector space over F . In particular, F 0 = f0g. 4. C is a vector space over R. 5. C is also a vector space over C. But, the vector spaces C (over R) and C (over C) are two different vector spaces. 6. Matrix spaces: Given any field F , the set of m×n matrices (this means m rows and n columns) with elements in F , denoted Mmn(F ), is a vector space over F where the addition and scalar multiplication are defined entry-wise. (Notation: the set of square n × n matrices with entries in F is denoted Mn(F ) rather than Mnn(F ).) 7. Sequence spaces: In these examples, the addition and scalar multiplication are defined term- by-term, i.e. (x1; x2; x3; :::)+(y1; y2; y3; :::) = (x1+y1; x2+y2; x3+y3; :::) and c(x1; x2; :::) = (cx1; cx2; :::): (a) Given any field F , the set F N of infinite sequences of elements of F is a vector space over F (where the addition and scalar multiplication are done term-by-term). (b) The set F 1 of infinite sequences where all but finitely many elements of the sequence are 0 also forms a vector space over F , with the same operations. Here, some care needs to be taken to ensure that addition is closed. (c) The set of convergent sequences of real numbers forms a vector space over R. 8. Function spaces: For all these spaces of functions, the addition is described by (f + g)(x) = f(x) + g(x) and the scalar multiplication is (cf)(x) = c · f(x) (a) Polynomials: Given any field F , the set F [x] of polynomials whose coefficients are in F is a vector space over F , where the addition and scalar multiplication come from usual addition and scalar multiplication of functions. (b) Continuous functions: The set C(R; R) of continuous functions from R to R is a vector space over R. (c) Differentiable functions: The set of differentiable functions from R to R is a vector space over R is a vector space over R. (d) Analytic functions: The set of analytic functions from R to R is a vector space over R is a vector space over R (a function is analytic if it can be written as a power series which converges everywhere). 2 R 1 2 (e) L functions: The set of \measurable" functions from R to R satsfying −∞ jf(x)j dx < 1 is a vector space over R (loosely speaking, a function is measurable if for every a R b and b in R, a f(x) dx exists... all continuous and piecewise-continuous functions are measurable)..