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lecture notes/section1fields-vector-spaces-subspaces-and-bases-pyu0ds4o.pdf 1 Fields, vector spaces, subspaces and bases Usually when you’ve used vectors, their elements have been Real. But we can be more general by allowing the elements to take values in a field. Other examples of fields include the complex numbers C, and integers modulo n (more about this soon). We want to consider vectors which can take values in any field, so we better start by defining what a field is. Definition 1.1 - a field. A field is a set with the operators addition (+) and multiplication (·). For a, b, c ∈ F , these operators satisfy: F1) Commutativity: a+ b = b+ a and a · b = b · a. F2) Associativity: a+ (b+ c) = (a+ b) + c and a · (b · c) = (a · b) · c. F3) Distributivity: a · (b+ c) = a · b+ a · c. F4) Identities: there are elements 0 and 1 such that a+ 0 = a and a · 1 = a. F5) Inverses: for every a ∈ F , there is an element −a ∈ F such that a+ (−a) = 0 and for every non-zero b ∈ F , there is an element b−1 ∈ F such that b−1 · b = b · b−1 = 1. You should check that the real numbers and complex numbers satisfy these axioms. We’ll also consider another example, the field of two numbers F2 = {0̄, 1̄}. When you define a field you must define what addition and multiplication means. We define it to mean 0̄ + 0̄ = 0̄, 0̄ + 1̄ = 1̄ + 0̄ = 1̄ and 1̄ + 1̄ = 0̄ . 0̄ · 0̄ = 0̄, 0̄ · 1̄ = 1̄ · 0̄ = 0̄ and 1̄ · 1̄ = 1̄ Note: since 1̄ + 1̄ = 0̄, we have −1̄ = 1̄. These models bits on a computer. A vector in R3 looks something like (1.1, 3.14, 2.001). Similarly a vector in (F2) 3 has elements like (1̄, 0̄, 1̄). Roughly, a vector space is a bunch of vectors. If you add two of them together, their sum should still in the space, and if you multiply one with a number (from a field, think R), it should also still be in the space. Let’s formulate this now. Definition 1.2 - A vector space. A vector space V over a field F is a set with two operations: addition and multiplication by scalars. The operators must satisfy five axioms: V1) Closure: If u,v ∈ V, α ∈ F then u + v ∈ V and αu ∈ V . V2) Commutativity and associativity of addition: u + v = v + u and u + (v + w) = (u + v) + w for all u,v,w ∈ V . V3) Distributivity and compatibility of scalar multiplication: α(u + v) = αu + αv, (α+ β)u = αu + βv and (αβ)u = α(βu) for all u ∈ V and α, β ∈ F . V4) 1v = v. V5) There is a unique element, which we’ll call 0 ∈ V , such that 0v = 0. (Note that 0 and 0 are different things.) Using these axioms, we can derive some other statements. 0 + v = v + 0 = v + 0v = (1 + 0)v = 1v = v Let’s get philosophical: we just used our knowledge about addition on the reals (we know that 1 + 0 = 0) to figure out something about addition in vector spaces. Now define −u = (−1)u. Now u + (−u) = u + (−1)u = (1− 1)u = 0u = 0 . 1 Hence we can define subtraction as: u− v = u + (−1)u. We are using axiomatic reasoning here. People who do logic or philosophy love this stuff. You can also try to define quantum mechanics axiomatically (this works well) or economics (this works... less well). Examples. 1) Let V = Rn and u,v ∈ V . So u = (u1, u2, ..., un) ,v = (v1, v2, ..., vn) with ui ∈ R and vi ∈ R for all i = 1, ..., n. We define addition as: u + v = (u1 + v1, u2 + v2, ..., un + vn) . And define multiplication as: αu = (αu1, αu2, ..., αun) for α ∈ R. These are sensible definitions. 2) Let F(R) be the set of all functions between reals. That is, if f(x) ∈ F(R), then f : R→ R, (Like f(x) = sin(x)) addition and multiplication are simple: f1(x) + f2(x) and αf1(x). 3) The set of matrices Matm,n(F ) with m row and n columns with elements taking values in F . Addition and scalar multiplication are straightforward. It looks like example 1) with many more numbers. 1.1 Subspaces Definition 1.3 - A subspace. A subspace of V is a subset U which contains 0 and is closed under addition and scalar multiplication. More formally we require: 0 ∈ U and if u1,u2 ∈ U and α ∈ F , then u1 + u2 ∈ U and αu1 ∈ U . Examples 1) Let U ⊂ R3 be defined as U = {(x, y, z) ∈ R3 : x+ y + z = 0} The first question is always: is this a subspace?. Three things to check a) Zero element? Here, 0 = (0, 0, 0) and 0 + 0 + 0 = 0, so 0 ∈ U . 3 b) Addition? Let x1 = (x1, y1, z1) and x2 = (x2, y2, z2) belong to U . Hence x1 + y1 + z1 = 0 and x2 + y2 + z2 = 0. Then x1 + x2 = (x1 + x2, y1 + y2, z1 + z2) satisfies (x1 + x2) + (y1 + y2) + (z1 + z2) = x1 + y1 + z1 + x2 + y2 + z2 = 0. Hence x1 + x2 ∈ U .s 3 c) Scalar multiplication? Let x = (x, y, z) ∈ U , then αx = (αx, αy, αz). We find that αx+ αy + αz = α(x+ y + z) = 0 and so αx ∈ U . 3 So U is a subspace! Geometrically U is a plane through the origin of R3. 2) Let U ⊂ R3 be defined as U = {(x, y, z) ∈ R3 : x+ y + z = 1} Is U a subspace? If you suspect something isn’t a subspace, the quickest way to prove it is by finding a counter-example. In this case (x, y, z) = (1, 0, 0) and (x, y, z) = (0, 1, 0) are both in U . But if we add them together, this is not in U , since (1, 1, 0) does not add up to 1. So U is not a subspace. 3) Let C∞(R) be the set of smooth functions. Smooth means that they can be differentiated as many times as you like. Most functions you know are smooth. Is C∞(R) a subspace of 2 F(R)? Three things: a) f(x) = 0 is the zero element. You can differentiate this forever since d0/dx = 0. 3 b) If you can differentiate f, g ∈ C∞(R) you can differentiate f + g. It’s just (f + g)′ = f ′ + g′. And we know we can do second derivatives by the same argument. And third derivatives. And fourth... 3 c) If you can differentiate f ∈ C∞(R) you can differentiate αf . It’s (αf)′ = αf ′. 3 4) Let Pn be the set of all polynomial functions of order n. That is, f(x) ∈ P if f(x) = a0 + a1x+ ...+ anx n = n∑ m=0 amx m for ai ∈ R. Exercise: Is P a subspace of F(R). Guess what? You’ve got to check three things. Exercise: How about the set of polynomials with fixed degree Pn, so that n is fixed in the above definition. A way to define this is that an 6= 0, while am = 0 for m > m. Is this a subspace of P? So the set of symmetric matrices is a subspace of Matm,n. 5) A symmetric matrix is one whose elements aij satisfy aij = aji. Examples include ( 0 2 2 −1 ) and ( 2 −3 −3 1 ) . The zero matrix (where all elements are zero) is symmetric, as is the sum of two symmetric matrices and a symmetric matrix multiplied by a number. 1.2 Spanning sets, linear independence, bases and dimension We’ve been thinking of big vector spaces, then thinking of elements in the spaces. You can also think the other way around: if you have an element, is it part of a space? What is the space? If you add another element, does that change things? Learning about spanning sets and linear independence will help us answer these questions. Definition 1.4 - linear span. The linear span of a finite set of vectors S = {v1,v2, ...,vn} in a vector space V is the set of all linear combinations of the vectors. That is: span(S) = {α1v1 + α2v2 + ...+ αnvn : α1, α2, ..., αn ∈ F} . (1) (Remember, F is just a field like R,C or F2) Example. If S contains just one vector with one element, perhaps a complex number, like S = {(1.34 + 2.1i)}, then V = F = C. So span(S) = {(1.34 + 2.1i)α : α ∈ C} = C. Intuitively, the span generates sets from elements of a set. We’ll now prove these sets are, in fact, subspaces. Proposition 1.5. span(S) is a subspace. Proof. Three things: 1) Zero element. 0 ∈ span(S) since 0 = 0v1 + ...+ 0vn. 3 2) Addition. If u = α1v1 + α2v2 + ...+ αnvn ∈ span(S) and v = β1v1 + β2v2 + ...+ βnvn ∈ span(S) then u + v = (α1 + β1)v1 + (α2 + β2)v2 + ...+ (αn + βn)vn, which is also in span(S). 3 3) Scalar multiplication. If u = α1v1 + α2v2 + ...+ αnvn ∈ span(S) then βu = βα1v1 + βα2v2 + ...+ βαnvn which is in span(S)