Vector Spaces Definition. A vector space (over R)isasetV with ◦ Let U and V be two sets. The cartesian adistinguishedelement0 ∈ V and two functions product U × V is the set of all ordered pairs of the form (a, b) where a ∈ A and b ∈ B. a : V × V V,denotedby a(u, v)=u + v, ◦ Example: if U = {1, 3} and V = {1, 3, 4},then and U×V = {(1, 1), (1, 3), (1, 4), (3, 1), (3, 3), (3, 4)}. → m : R × V V,denotedbym(c, v)=c.v = cv, ◦ We write U2 = U × U, U3 = U × U × U and so on. such that the following axioms hold for all u, v, w ∈ V →and for all c, d ∈ R: ◦ We have been using this notation already ◦ u + v = v + u, when we write R2 or R3. ◦ (u + v)+w = u +(v + w), ◦ Vector addition on Rn define a function Rn Rn Rn a : × .Insteadofwriting ◦ u + 0 = u, 3 4 7 ◦ u ∈ V (−u) ∈ V = For all ,thereexists such a→ 2 , −8 −6 , that u +(−u)=0, !" # " #$ " # we write, ◦ c(u + v)=cu + cv, 3 4 7 ◦ (c + d)u = cu + du, + = . 2 −8 −6 ◦ c(du)=(cd)u " # " # " # , for convenience of notation. ◦ 1u = u. Some nomenclature and examples ◦ Let V be the set of all doubly infinite sequences of real numbers of the form In the contex of the above definition: (··· , y−2, y−1, y0, y1, y2, ···) ◦ elements of V are called vectors (of V). where each yj ∈ R.Theadditionandscalar ◦ the functions a and m are called addition multiplication are defined componentwise. and scalar multiplication. ◦ Let V be the set of all infinitely differentiable ◦ Functions like a are often called binary functions defined on R.Here0V : R R is operations. the function ◦ The element 0 ∈ V is called the zero vector 0V(x)=0 for all x ∈ R. → of V.Weshallwrite0 = 0V if there is chance of confusion. If f, g ∈ V and c ∈ R,then(f + g) and cf are defined by First examples: Next we give some examples of vector spaces. We do not specify the 0 vector and (f + g)(x)=f(x)+g(x) and (cf)(x)=cf(x). the notions of addition and scalar multiplication when they are obvious. ◦ Let V = Rn. ◦ Let V = Mm,n = the set of all m × n matrices. ◦ Let V = Pn = set of polynomials of degree at most n. ◦ Let V = {0} Exercise. Let (V, +,.,0) be a vector space. Show that for all u ∈ V and for all c ∈ R,wehave ◦ 0.u = u (Here 0 means 0R). ◦ c.0 = 0 (Here 0 means 0V). ◦ −u =(−1).u. Subspaces ◦ The span of S, denoted by span(S) means the set of all linear combinations of elements Let V be a vector space. of S. Definition. AsubsetH of V is called a subspace of Examples of subspaces: V if H is a vector space with the zero vector, addition and scalar multiplication induced from ◦ – Let V be a vector space and let S be a V. subset of V. – One verifies that span(S) is a subspace Observe: AsubsetH of V is a subspace if and of V. only if: – Infact, span(S) is the smallest subspace ◦ 0V ∈ H, of V that contains S. – ◦ u, v ∈ H implies u + v ∈ H, If H = span(S), we say that S spans H or that H is spanned by S. ◦ u ∈ H and c ∈ R implies c.u ∈ H, ◦ Let A be an n × m matrix whose columns ⃗ ⃗ where + and . denotes the operations on V. are a1, ···am.Recallthatwedefined Definition: Let V be a vector space. Let S be a ⃗ Rm ⃗ ⃗ subset of V. Nul(A)={x ∈ : Ax = 0} ◦ S and A linear combination of elements of ⃗ ⃗ means an element of V of the form Col(A)=span{a1, ··· , am}. One verifies that (c1v1 + ···+ cnvn) – Nul(A) is a subspace of Rm where each cj ∈ R and each vj ∈ S. – Col(A) is a subspace of Rn. Exercise: detecting subspaces Exercise: membership in span Linear Transformations 4 Definition. Let V and W be vector spaces. A ◦ Let L : M22 R be the map linear transformation T : V W is a function from V to W such that a → ab b T(u + v)=T(u)+T(v) →and T(cu)=cT(u) L cd = ⎡c⎤ . !" #$ d for all u, v ∈ V and for all c ∈ R. ⎢ ⎥ ⎣ ⎦ Examples: The maps defined below are all linear. ◦ L : Mm,n Mn,m defined by The verifications are easy in each case. L(A)=AT . ◦ An n × m matrix A defines a linear map → n m T LA : R R defined by Here A denotes transpose of A. R LA(⃗v)=A⃗v. ◦ Let C ( ) be the vector space of all → infinitely differentiable functions on R. ∞ All linear maps from Rn to Rm has this form. Define D : C (R) C (R) by R ∞ ∞ ◦ Let Tr : Mn be the function D(f)=f′ (the derivative of f) → Tr(( aij)) = a11 + a22 + ···+ ann. → ◦ Let C([0, 2]) be the space of all continuous R In other words, Tr(A) is the sum of the functions on [0, 2].Defineev1 : C([0, 2]) diagonal entries of A.WesayTr(A) is the by ev (f)=f(1). trace of A. 1 → Kernel and range Let T : V W be a linear transformation (also ◦ If T : V W is a linear map, then verify that called a linear map). ker(T) is a subspace of V and range(T) is a subspace of W. ◦ V is→ called the domain of T. → ◦ Let A be an n × m matrix. Recall that A ◦ W is called the codomain of T. n m defines a linear map LA : R R defined ◦ The range of T, denoted range(T), means the by LA(⃗v)=A⃗v.Verifythat set of all vectors of W that can be written in → the form T(v) for some v ∈ V. ker(LA)=nul(A) and range(LA)=col(A). ◦ The kernel of T,denotedbyker(T), means the set of all v ∈ V such that T(v)=0..