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Chapter 3 Direct Sums, Affine Maps, the Dual Space, Duality

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Chapter 3 Direct Sums, Affine Maps, the Dual Space, Duality Chapter 3 Direct Sums, Affine Maps, The Dual Space, Duality 3.1 Direct Products, Sums, and Direct Sums There are some useful ways of forming new vector spaces from older ones. Definition 3.1. Given p 2vectorspacesE ,...,E , ≥ 1 p the product F = E1 Ep can be made into a vector space by defining addition⇥···⇥ and scalar multiplication as follows: (u1,...,up)+(v1,...,vp)=(u1 + v1,...,up + vp) λ(u1,...,up)=(λu1,...,λup), for all ui,vi Ei and all λ R. 2 2 With the above addition and multiplication, the vector space F = E E is called the direct product of 1 ⇥···⇥ p the vector spaces E1,...,Ep. 151 152 CHAPTER 3. DIRECT SUMS, AFFINE MAPS, THE DUAL SPACE, DUALITY The projection maps pr : E E E given by i 1 ⇥···⇥ p ! i pri(u1,...,up)=ui are clearly linear. Similarly, the maps in : E E E given by i i ! 1 ⇥···⇥ p ini(ui)=(0,...,0,ui, 0,...,0) are injective and linear. It can be shown (using bases) that dim(E E )=dim(E )+ +dim(E ). 1 ⇥···⇥ p 1 ··· p Let us now consider a vector space E and p subspaces U1,...,Up of E. We have a map a: U U E 1 ⇥···⇥ p ! given by a(u ,...,u )=u + + u , 1 p 1 ··· p with u U for i =1,...,p. i 2 i 3.1. DIRECT PRODUCTS, SUMS, AND DIRECT SUMS 153 It is clear that this map is linear, and so its image is a subspace of E denoted by U + + U 1 ··· p and called the sum of the subspaces U1,...,Up. By definition, U + + U = u + + u u U , 1 i p , 1 ··· p { 1 ··· p | i 2 i } and it is immediately verified that U + + U is the 1 ··· p smallest subspace of E containing U1,...,Up. If the map a is injective, then Ker a = (0,...,0 ) , { } p which means that if ui Ui for i =1,...,p and if 2 | {z } u + + u =0 1 ··· p then u1 =0,...,up =0. In this case, every u U1 + + Up has a unique ex- pression as a sum 2 ··· u = u + + u , 1 ··· p with u U ,fori =1,...,p. i 2 i 154 CHAPTER 3. DIRECT SUMS, AFFINE MAPS, THE DUAL SPACE, DUALITY It is also clear that for any p nonzero vectors u U , i 2 i u1,...,up are linearly independent. Definition 3.2. For any vector space E and any p 2 ≥ subspaces U1,...,Up of E,ifthemapa defined above is injective, then the sum U1 + + Up is called a direct sum and it is denoted by ··· U U . 1 ⊕···⊕ p The space E is the direct sum of the subspaces Ui if E = U U . 1 ⊕···⊕ p Observe that when the map a is injective, then it is a linear isomorphism between U1 Up and U U . ⇥···⇥ 1 ⊕···⊕ p The di↵erence is that U U is defined even if 1 ⇥···⇥ p the spaces Ui are not assumed to be subspaces of some common space. There are natural injections from each Ui to E denoted by in : U E. i i ! 3.1. DIRECT PRODUCTS, SUMS, AND DIRECT SUMS 155 Now, if p =2,itiseasytodeterminethekernelofthe map a: U U E.Wehave 1 ⇥ 2 ! a(u ,u )=u + u =0i↵u = u ,u U ,u U , 1 2 1 2 1 − 2 1 2 1 2 2 2 which implies that Ker a = (u, u) u U U . { − | 2 1 \ 2} Now, U U is a subspace of E and the linear map 1 \ 2 u (u, u)isclearlyanisomorphismbetweenU1 U2 and7! Ker a−,soKera is isomorphic to U U . \ 1 \ 2 As a consequence, we get the following result: Proposition 3.1. Given any vector space E and any two subspaces U1 and U2, the sum U1 + U2 is a direct sum i↵ U U =(0). 1 \ 2 156 CHAPTER 3. DIRECT SUMS, AFFINE MAPS, THE DUAL SPACE, DUALITY Recall that an n n matrix A M is symmetric if ⇥ 2 n A> = A, skew -symmetric if A> = A.Itisclearthat − S(n)= A M A> = A { 2 n | } Skew(n)= A M A> = A { 2 n | − } are subspaces of M ,andthatS(n) Skew(n)=(0). n \ Observe that for any matrix A M ,thematrixH(A)= 2 n (A + A>)/2issymmetricandthematrix S(A)=(A A>)/2isskew-symmetric.Since − A + A> A A> A = H(A)+S(A)= + − , 2 2 we have the direct sum M = S(n) Skew(n). n ⊕ 3.1. DIRECT PRODUCTS, SUMS, AND DIRECT SUMS 157 Proposition 3.2. Given any vector space E and any p 2 subspaces U1,...,Up, the following properties are≥ equivalent: (1) The sum U + + U is a direct sum. 1 ··· p (2) We have p U U =(0),i=1,...,p. i \ j ✓ j=1,j=i ◆ X6 (3) We have i 1 − U U =(0),i=2,...,p. i \ j j=1 ✓ X ◆ The isomorphism U U U U implies 1 ⇥···⇥ p ⇡ 1 ⊕···⊕ p Proposition 3.3. If E is any vector space, for any (finite-dimensional) subspaces U1,..., Up of E, we have dim(U U )=dim(U )+ +dim(U ). 1 ⊕···⊕ p 1 ··· p 158 CHAPTER 3. DIRECT SUMS, AFFINE MAPS, THE DUAL SPACE, DUALITY If E is a direct sum E = U U , 1 ⊕···⊕ p since every u E can be written in a unique way as 2 u = u + + u 1 ··· p for some ui Ui for i =1...,p,wecandefinethemaps ⇡ : E U ,called2 projections,by i ! i ⇡ (u)=⇡ (u + + u )=u . i i 1 ··· p i It is easy to check that these maps are linear and satisfy the following properties: ⇡i if i = j ⇡j ⇡i = ◦ 0ifi = j, ( 6 ⇡ + + ⇡ =id . 1 ··· p E 3.1. DIRECT PRODUCTS, SUMS, AND DIRECT SUMS 159 For example, in the case of the direct sum M = S(n) Skew(n), n ⊕ the projection onto S(n)isgivenby A + A ⇡ (A)=H(A)= >, 1 2 and the projection onto Skew(n)isgivenby A A> ⇡ (A)=S(A)= − . 2 2 Clearly, H(A)+S(A)=A, H(H(A)) = H(A), S(S(A)) = S(A), and H(S(A)) = S(H(A)) = 0. Afunctionf such that f f = f is said to be idempotent. ◦ Thus, the projections ⇡i are idempotent. Conversely, the following proposition can be shown: 160 CHAPTER 3. DIRECT SUMS, AFFINE MAPS, THE DUAL SPACE, DUALITY Proposition 3.4. Let E be a vector space. For any p 2 linear maps f : E E, if ≥ i ! fi if i = j fj fi = ◦ 0 if i = j, ( 6 f + + f =id , 1 ··· p E then if we let Ui = fi(E), we have a direct sum E = U U . 1 ⊕···⊕ p We also have the following proposition characterizing idem- potent linear maps whose proof is also left as an exercise. Proposition 3.5. For every vector space E, if f : E E is an idempotent linear map, i.e., f f = f, then we! have a direct sum ◦ E =Kerf Im f, ⊕ so that f is the projection onto its image Im f. We are now ready to prove a very crucial result relating the rank and the dimension of the kernel of a linear map. 3.1. DIRECT PRODUCTS, SUMS, AND DIRECT SUMS 161 Theorem 3.6. Let f : E F be a linear map. For ! any choice of a basis (f1,...,fr) of Im f, let (u1,...,ur) be any vectors in E such that fi = f(ui), for i = 1,...,r.Ifs:Imf E is the unique linear map de- ! fined by s(fi)=ui, for i =1,...,r, then s is injective, f s =id, and we have a direct sum ◦ E =Kerf Im s ⊕ as illustrated by the following diagram: f / Ker f / E =Kerf Im s Im f F. ⊕ o s ✓ As a consequence, dim(E)=dim(Kerf)+dim(Imf)=dim(Kerf)+rk(f). Remark: The dimension dim(Ker f)ofthekernelofa linear map f is often called the nullity of f. We now derive some important results using Theorem 3.6. 162 CHAPTER 3. DIRECT SUMS, AFFINE MAPS, THE DUAL SPACE, DUALITY Proposition 3.7. Given a vector space E, if U and V are any two subspaces of E, then dim(U)+dim(V )=dim(U + V )+dim(U V ), \ an equation known as Grassmann’s relation. The Grassmann relation can be very useful to figure out whether two subspace have a nontrivial intersection in spaces of dimension > 3. For example, it is easy to see that in R5,therearesub- spaces U and V with dim(U)=3anddim(V )=2such that U V =(0). \ However, we can show that if dim(U)=3anddim(V )= 3, then dim(U V ) 1. \ ≥ As another consequence of Proposition 3.7, if U and V are two hyperplanes in a vector space of dimension n,so that dim(U)=n 1anddim(V )=n 1, we have − − dim(U V ) n 2, \ ≥ − and so, if U = V ,then 6 dim(U V )=n 2. \ − 3.1. DIRECT PRODUCTS, SUMS, AND DIRECT SUMS 163 Proposition 3.8. If U1,...,Up are any subspaces of a finite dimensional vector space E, then dim(U + + U ) dim(U )+ +dim(U ), 1 ··· p 1 ··· p and dim(U + + U )=dim(U )+ +dim(U ) 1 ··· p 1 ··· p i↵the U s form a direct sum U U . i 1 ⊕···⊕ p Another important corollary of Theorem 3.6 is the fol- lowing result: Proposition 3.9. Let E and F be two vector spaces with the same finite dimension dim(E)=dim(F )= n.
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