Data for a vector space

Transcription

1 Data for a vector space Aim lecture: We recall the notion of a vector space which provides the context for describing linear phenomena. Throughout the rest of these lectures, F denotes a field. Defn A vector space over F (or F-space) consists of a set V of elements called vectors and two maps 1 addition: V V V : (v, v ) v + v 2 scalar multiplication: F V V : (α, v) αv such that the following axioms hold Daniel Chan (UNSW) Lecture 7: Vector spaces Semester / 12

2 Axioms for a vector space Axioms For all v, v V, β, β F we have 1 (V, +) is an abelian group. 2 (ββ )v = (β(β v)) 3 1v = v 4 (β + β )v = βv + β v 5 β(v + v ) = βv + βv Rem It is a good ex to show that (assuming 1) & 4)), axioms 2) & 3) above amount to saying that the group F acts on V. Daniel Chan (UNSW) Lecture 7: Vector spaces Semester / 12

3 Simple examples of vector spaces E.g. 1 F n is a vector space over F E.g. 2 Geometric vectors in 3-dim space is a vector space over F = R. E.g. 3 M mn (F) is an F-space. Daniel Chan (UNSW) Lecture 7: Vector spaces Semester / 12

4 More examples E.g. 4 F[x] = the set of polynomials in the indet x & co-eff from F is an F-space. E.g. 5 For a set X, the set of fns from X F, denoted Fun(X, F) is a vector space with pointwise addn & scalar multn. i.e. For f, g Fun(X, F), β F, x X we have (f + g)(x) = f (x) + g(x), (βf )(x) = β(f (x)). E.g. 6 0 is a vector space over any F. Daniel Chan (UNSW) Lecture 7: Vector spaces Semester / 12

5 Basic properties You know the following (& should prove yourself if you ve forgotten how) Prop For any F-space V and v V, β F 1 The zero vector & negative of a vector are unique. 2 β0 = 0 3 0v = 0 4 ( 1)v = v 5 βv = 0 = β = 0 or v = 0. Daniel Chan (UNSW) Lecture 7: Vector spaces Semester / 12

6 Vector arithmetic One of the key points of the axioms of a vector space V, is that for v 1,..., v n V, β 1,..., β n F one can form the vector β 1 v 1 + β 2 v β n v n. Such a vector or expression is called a linear combination of v 1,..., v n. Furthermore, axioms show such linear combinations can be manipulated arithmetically, in the way one handles n-tuples of reals. E.g. Daniel Chan (UNSW) Lecture 7: Vector spaces Semester / 12

7 Subspaces To get more examples of vector spaces Prop-Defn Let V = F-space & W be a non-empty subset. Then the following are equivalent. 1 W is closed under addn & scalar multn i.e. for w, w W, β F we have w + w W and βw W. 2 For any w, w W, β F we have βw + w W. 3 The linear combination of any set of elements of W lies in W. If these conditions hold, then the addn & scalar multn laws on V restrict to addn & scalar multn laws on W making W an F-space too. In this case we say W is a subspace of V & write W V. E.g. Any vector space V has two trivial subspaces Daniel Chan (UNSW) Lecture 7: Vector spaces Semester / 12

8 Examples of subspaces E.g. 1 If one identifies R[x] with the set of real polynomial fns, then R[x] is a subspace of Fun(R, R). E.g. 2 Let I R be an interval & C 0 (I ) denote the set of continuous functions on I. Then C 0 (I ) is a subspace of Fun(I, R) since E.g.3 Let i be a positive integer or & I R be an open interval. Let C i (I ) denote the set of i-times continuously differentiable fns on I. Then C i (I ) is a subspace of C j (I ) for j < i. E.g. 4 For each d N, the subset F[x] d F[x] of polynomials of degree d is a subspace. Daniel Chan (UNSW) Lecture 7: Vector spaces Semester / 12

9 Some basic results about subspaces Prop 1 If U V, V W then U W. Sim if U W, V W, U V then U V. 2 If v is a vector of an F-space then F v = {βv β F} is a subspace. 3 Let V be a vector space and V 1,..., V r be subspaces. Then r i=1 V i is also a subspace of V. 4 With the above notn, r i=1 V i = V V r = { i v i v i V i } is a subspace of V called the sum of the V i. Proof. All follow from checking closure axioms. We prove 4) as an example. Consider w, w V i, β F. Then we may write w = i v i, w = i v i for some v i, v i V i. Closure axioms ensure βv i + v i V i so βw + w = i (βv i + v i) i V i. Daniel Chan (UNSW) Lecture 7: Vector spaces Semester / 12

10 Lines and Planes Let v, w R 3 be non-parallel vectors. Recall from MATH1141/MATH1151 Fact R v is the line in R 3 with parametric equation Fact R v + R w is the plane in R 3 with parametric equation Daniel Chan (UNSW) Lecture 7: Vector spaces Semester / 12

11 Example of sums E.g. Let W = R(1, 1, 0) T + R(0, 2, 1) T, W = R(1, 1, 1) T. Does W + W = R 3? Daniel Chan (UNSW) Lecture 7: Vector spaces Semester / 12

12 Example of intersections E.g. Find the intersection of the planes W = R(1, 2, 1) T + R(1, 1, 1) T, W = R(0, 2, 1) + R(1, 1, 0) T. Daniel Chan (UNSW) Lecture 7: Vector spaces Semester / 12