Exercises 3F [[3F-1]] Explain why each linear functional is surjective or is the zero map. [[3F-2]]  Give three distinct examples of linear functionals on 𝐑[0,1]. [[3F-3]]  Suppose 𝑉 is finite-dimensional and 𝑣 ∈ 𝑉 with 𝑣 ≠ 0. Prove that there exists 𝜑 ∈ 𝑉′ such that 𝜑(𝑣) = 1. [[3F-4]]  Suppose 𝑉 is finite-dimensional and 𝑈 is a subspace of 𝑉 such that 𝑈 ≠ 𝑉. Provethatthereexists𝜑 ∈ 𝑉′ suchthat𝜑(𝑢) = 0forevery𝑢 ∈ 𝑈but𝜑 ≠ 0. [[3F-5]]  Suppose 𝑇 ∈ L(𝑉, 𝑊) and 𝑤1, ..., 𝑤𝑚 is a basis of range 𝑇. Hence for each 𝑣 ∈ 𝑉, there exist unique numbers 𝜑1(𝑣), ..., 𝜑𝑚(𝑣) such that 𝑇𝑣 = 𝜑1(𝑣)𝑤1 + ⋯ + 𝜑𝑚(𝑣)𝑤𝑚, thus defining functions 𝜑1, ..., 𝜑𝑚 from 𝑉 to 𝐅. Show that each of the functions 𝜑1, ..., 𝜑𝑚 is a linear functional on 𝑉. [[3F-6]]  Suppose 𝜑, 𝛽 ∈ 𝑉′. Prove that null 𝜑 ⊆ null 𝛽 if and only if there exists 𝑐 ∈ 𝐅 such that 𝛽 = 𝑐𝜑. [[3F-7]]  Suppose that 𝑉1,...,𝑉𝑚 are vector spaces. Prove that (𝑉1 ×⋯×𝑉𝑚)′ and 𝑉1′ × ⋯ × 𝑉𝑚′ are isomorphic vector spaces. [[3F-8]]  Suppose 𝑣1, ..., 𝑣𝑛 is a basis of 𝑉 and 𝜑1, ..., 𝜑𝑛 is the dual basis of 𝑉′. Define Γ∶ 𝑉 → 𝐅𝑛 and Λ∶ 𝐅𝑛 → 𝑉 by Γ(𝑣) = (𝜑1(𝑣), ..., 𝜑𝑛(𝑣)) and Λ(𝑎1, ..., 𝑎𝑛) = 𝑎1𝑣1 + ⋯ + 𝑎𝑛𝑣𝑛. Explain why Γ and Λ are inverses of each other. [[3F-9]]  Suppose 𝑚 is a positive integer. Show that the dual basis of the basis 1,𝑥,...,𝑥𝑚 of 𝒫𝑚(𝐑) is 𝜑0,𝜑1,...,𝜑𝑚, where 𝜑𝑘(𝑝) = 𝑝(𝑘)(0)/ 𝑘! Here 𝑝(𝑘) denotes the 𝑘th derivative of 𝑝, with the understanding that the 0th derivative of 𝑝 is 𝑝. [[3F-10]]  Suppose 𝑚 is a positive integer. (a) Show that 1,𝑥 − 5,...,(𝑥 − 5)𝑚 is a basis of 𝒫𝑚(𝐑). (b) What is the dual basis of the basis in (a)? [[3F-11]]  Suppose 𝑣1, ..., 𝑣𝑛 is a basis of 𝑉 and 𝜑1, ..., 𝜑𝑛 is the corresponding dual basis of 𝑉′. Suppose 𝜓 ∈ 𝑉′. Prove that 𝜓 = 𝜓(𝑣1)𝜑1 + ⋯ + 𝜓(𝑣𝑛)𝜑𝑛. Prove that 𝜓 = 𝜓(𝑣1)𝜑1 + ⋯ + 𝜓(𝑣𝑛)𝜑𝑛. [[3F-12]]  Suppose 𝑆,𝑇 ∈ L(𝑉,𝑊). (a) Prove that (𝑆+𝑇)′ =𝑆′ +𝑇′. (b) Prove that (𝜆𝑇)′ = 𝜆𝑇′ for all 𝜆 ∈ 𝐅. This exercise asks you to verify (a) and (b) in 3.120. [[3F-13]]  Show that the dual map of the identity operator on 𝑉 is the identity operator on 𝑉′. [[3F-14]]  Define𝑇∶𝐑3→𝐑2by 𝑇(𝑥,𝑦,𝑧) = (4𝑥+5𝑦+6𝑧,7𝑥+8𝑦+9𝑧). Suppose 𝜑1,𝜑2 denotes the dual basis of the standard basis of 𝐑2 and 𝜓1, 𝜓2, 𝜓3 denotes the dual basis of the standard basis of 𝐑3. (a) Describe the linear functionals 𝑇′(𝜑1) and 𝑇′(𝜑2). (b) Write 𝑇′(𝜑1) and 𝑇′(𝜑2) as linear combinations of 𝜓1, 𝜓2, 𝜓3. [[3F-15]]  Define𝑇∶𝒫(𝐑)→𝒫(𝐑)by (𝑇𝑝)(𝑥) = 𝑥2𝑝(𝑥) + 𝑝′′(𝑥) for each 𝑥 ∈ 𝐑. 1. (a)  Suppose 𝜑 ∈ 𝒫(𝐑)′ is defined by 𝜑(𝑝) = 𝑝′(4). Describe the linear functional 𝑇′(𝜑) on 𝒫(𝐑). 2. (b)  Suppose 𝜑 ∈ 𝒫(𝐑)′ is defined by 𝜑(𝑝) = ∫1 𝑝. Evaluate (𝑇′(𝜑))(𝑥3). 0 [[3F-16]]  Suppose 𝑊 is finite-dimensional and 𝑇 ∈ L(𝑉, 𝑊). Prove that 𝑇′ = 0 ⟺ 𝑇 = 0. [[3F-17]]  Suppose 𝑉 and 𝑊 are finite-dimensional and 𝑇 ∈ L(𝑉, 𝑊). Prove that 𝑇 is invertible if and only if 𝑇′ ∈ L(𝑊′, 𝑉′) is invertible. [[3F-18]]  Suppose 𝑉 and 𝑊 are finite-dimensional. Prove that the map that takes 𝑇 ∈ L(𝑉, 𝑊) to 𝑇′ ∈ L(𝑊′, 𝑉′) is an isomorphism of L(𝑉, 𝑊) onto L(𝑊′, 𝑉′). [[3F-19]]  Suppose 𝑈 ⊆ 𝑉. Explain why 𝑈0 = {𝜑 ∈ 𝑉′ ∶𝑈 ⊆ null𝜑}. [[3F-20]]  Suppose 𝑉 is finite-dimensional and 𝑈 is a subspace of 𝑉. Show that 𝑈 = {𝑣 ∈ 𝑉 ∶ 𝜑(𝑣) = 0 for every 𝜑 ∈ 𝑈0}. [[3F-21]]  Suppose 𝑉 is finite-dimensional and 𝑈 and 𝑊 are subspaces of 𝑉. (a) Prove that 𝑊0 ⊆𝑈0 if and only if 𝑈⊆𝑊. (b) Prove that 𝑊0 =𝑈0 if and only if 𝑈=𝑊. [[3F-22]]  Suppose 𝑉 is finite-dimensional and 𝑈 and 𝑊 are subspaces of 𝑉. (a) Show that (𝑈 + 𝑊)0 = 𝑈0 ∩ 𝑊0. (b) Show that (𝑈 ∩ 𝑊)0 = 𝑈0 + 𝑊0. [[3F-23]]  Suppose 𝑉 is finite-dimensional and 𝜑1, ..., 𝜑𝑚 ∈ 𝑉′. Prove that the following three sets are equal to each other. (a) span(𝜑1, ..., 𝜑𝑚) (b) ((null𝜑 ) ∩ ⋯ ∩ (null𝜑 ))0 (c) {𝜑 ∈ 𝑉′ ∶ (null𝜑1) ∩ ⋯ ∩ (null𝜑𝑚) ⊆ null𝜑} [[3F-24]]  Suppose 𝑉 is finite-dimensional and 𝑣1, ..., 𝑣𝑚 ∈ 𝑉. Define a linear map Γ∶𝑉′ →𝐅𝑚 byΓ(𝜑)=(𝜑(𝑣1),...,𝜑(𝑣𝑚)). (a) Prove that 𝑣1, ..., 𝑣𝑚 spans 𝑉 if and only if Γ is injective. (b) Prove that 𝑣1, ..., 𝑣𝑚 is linearly independent if and only if Γ is surjective. [[3F-25]]  Suppose 𝑉 is finite-dimensional and 𝜑1, ..., 𝜑𝑚 ∈ 𝑉′. Define a linear map Γ∶𝑉→𝐅𝑚 byΓ(𝑣)=(𝜑1(𝑣),...,𝜑𝑚(𝑣)). (a) Prove that 𝜑1, ..., 𝜑𝑚 spans 𝑉′ if and only if Γ is injective. (b) Prove that 𝜑1, ..., 𝜑𝑚 is linearly independent if and only if Γ is surjective. [[3F-26]]  Suppose 𝑉 is finite-dimensional and Ω is a subspace of 𝑉′. Prove that Ω = {𝑣 ∈ 𝑉 ∶ 𝜑(𝑣) = 0 for every 𝜑 ∈ Ω}0. [[3F-27]]  Suppose 𝑇 ∈ L(𝒫5(𝐑)) and null𝑇′ = span(𝜑), where 𝜑 is the linear functional on 𝒫5(𝐑) defined by 𝜑(𝑝) = 𝑝(8). Prove that range 𝑇 = {𝑝 ∈ 𝒫5(𝐑) ∶ 𝑝(8) = 0}. [[3F-28]]  Suppose 𝑉 is finite-dimensional and 𝜑1, ..., 𝜑𝑚 is a linearly independent list in 𝑉′. Prove that dim((null 𝜑1) ∩ ⋯ ∩ (null 𝜑𝑚)) = (dim 𝑉) − 𝑚. [[3F-29]]  Suppose 𝑉 and 𝑊 are finite-dimensional and 𝑇 ∈ L(𝑉, 𝑊). (a) Prove that if 𝜑 ∈ 𝑊′ and null 𝑇′ = span(𝜑), then range 𝑇 = null 𝜑. (b) Prove that if 𝜓 ∈ 𝑉′ and range 𝑇′ = span(𝜓), then null 𝑇 = null 𝜓. [[3F-30]]  Suppose 𝑉 is finite-dimensional and 𝜑1, ..., 𝜑𝑛 is a basis of 𝑉′. Show that there exists a basis of 𝑉 whose dual basis is 𝜑1, ..., 𝜑𝑛. [[3F-31]]  Suppose 𝑈 is a subspace of 𝑉. Let 𝑖 ∶ 𝑈 → 𝑉 be the inclusion map defined by 𝑖(𝑢) = 𝑢. Thus 𝑖′ ∈ L(𝑉′, 𝑈′). 1. (a)  Showthatnull𝑖′ =𝑈0. 2. (b)  Prove that if 𝑉 is finite-dimensional, then range 𝑖′ = 𝑈′. 3. (c)  Prove that if 𝑉 is finite-dimensional, then 𝑖′̃ is an isomorphism from 𝑉′/𝑈0 onto 𝑈′. The isomorphism in (c) is natural in that it does not depend on a choice of basis in either vector space. [[3F-32]] The double dual space of 𝑉, denoted by 𝑉′′, is defined to be the dual space of 𝑉′. In other words, 𝑉′′ = (𝑉′)′. Define Λ∶ 𝑉 → 𝑉′′ by (Λ𝑣)(𝜑) = 𝜑(𝑣) for each 𝑣 ∈ 𝑉 and each 𝜑 ∈ 𝑉′. 1. (a)  Show that Λ is a linear map from 𝑉 to 𝑉′′. 2. (b)  Showthatif𝑇∈L(𝑉),then𝑇′′∘Λ=Λ∘𝑇,where𝑇′′ =(𝑇′)′. 3. (c)  Showthatif𝑉isfinite-dimensional,then Λ is an isomorphism from 𝑉 onto 𝑉′′. Suppose 𝑉 is finite-dimensional. Then 𝑉 and 𝑉′ are isomorphic, but finding an isomorphism from 𝑉 onto 𝑉′ generally requires choosing a basis of 𝑉. In contrast, the isomorphism Λ from 𝑉 onto 𝑉′′ does not require a choice of basis and thus is considered more natural. [[3F-33]] Suppose 𝑈is a subspace of 𝑉. Let 𝜋∶𝑉→𝑉/𝑈 be the usual quotient map. Thus 𝜋′ ∈ L((𝑉 𝑈)', 𝑉′). (a) Show that 𝜋′ is injective. (b) Show that range 𝜋′ = 𝑈0. (c) Conclude that 𝜋′ is an isomorphism from (𝑉/𝑈)′ onto 𝑈0. The isomorphism in (c) is natural in that it does not depend on a choice of basis in either vector space. In fact, there is no assumption here that any of these vector spaces are finite-dimensional. ---