Physics 3210 Spring 2008Problem Set 11. Let S be any set, and consider the collection V of all mappings f of S intoa field F. For any f, g ∈ V and α ∈ F, we define (f + g)(x) = f (x) + g(x)and (αf)(x) = αf(x) for every x ∈ S. Show that V together with theseoperations defines a vector space over F.2. Write the vector v = (1, −2, 5) as a linear combination of the vectorsx1= (1, 1, 1), x2= (1, 2, 3) and x3= (2, −1, 1).3. Determine whether or not the three vectors x1= (2, −1, 0), x2= (1, −1, 1)and x3= (0, 2, 3) form a basis for R3.4. Extend each of the following sets to a basis for the given space:(a) {(1, 1, 0 ), (2, −2, 0)} in R3.(b) {(1, 0, 0, 0), (1, 0, 1, 0), (1, 0, 0, 1)} in R4.(c) {(1, 1, 0, 0), (1, −1, 0, 0), (1, 0, 1, 0)} in R4.5. Show that the vectors u = (1 + i, 2i), v = (1, 1 + i) ∈ C2are linearlydependent over C, but linear ly independent over R.6. Show that no linearly independent set of vectors can contain the zerovector.7. Find the coordinates of the vector (3, 1, − 4) ∈ R3relative to the basisx1= (1, 1, 1), x2= (0, 1, 1) and x3= (0, 0, 1).8. Let φ : V → W be an isomo rphism of finite-dimensio nal vector spaces.Show that a set of vectors {φ(v1), . . . , φ(vn)} is linearly independent in Wif and only if the set {v1, . . . , vn} is linearly independent in V . Equiva-lently, you may replace “independent” by “dependent.”9. Let φ : V → W be a homomorphism of two vector spaces V and W .(a) Show that φ maps any subspace of V onto a subspace of W .(b) Let S′be a subspace of W , and define the set S = {x ∈ V : φ(x) ∈ S′}.Show that S is a subspace of V .10. Let V have basis x1, x2, . . . , xn, and let v1, v2, . . . , vnbe any n elementsin V . Define a mapping φ : V → V byφ nXi=1aixi!=nXi=1aiviwhere each ai∈ F.(a) Show that φ is a homomor phism.(b) When is φ an isomorphism?111. Let W1and W2be subspaces of R3defined by W1= {(x, y, z) : x = y = z}and W2= {(x, y, z) : x = 0}. Show that R3= W1⊕ W2.12. Let V = F [R] denote the space of all real-valued functions defined on Rwith a ddition and scalar multiplication defined as in Exercise 1. In otherwords, f ∈ F [R] mea ns f : R → R. Let W+and W−be the subsets of Vdefined by W+= {f ∈ V : f (−x) = f(x)} and W−= {f ∈ V : f (−x) =−f (x)}. In other words, W+is the subset of a ll even functions, and W−is the subset of all odd functions.(a) Show that W+and W−are subspaces of V .(b) Show that V = W+⊕ W−.13. Let x = (x1, x2) and y = (y1, y2) be vectors in R2, and define the mappingh· , ·i : R2→ R by hx, yi = x1y1− x1y2− x2y1+ 3x2y2. Show this definesan inner pr oduct on R2.14. Let x = (3, 4) ∈ R2, and evaluate kxk with resp e c t to the norm inducedby:(a) The standard inner product on R2.(b) The inner product defined in Exercise 13.15. Let V be an inner product space, and let x, y ∈ V .(a) Prove the parallelogram law:kx + yk2+ kx − yk2= 2 kxk2+ 2 kyk2.(The geometric meaning of this equation is that the sum of the squaresof the diagonals of a parallelog ram is equal to the sum of the squar esof the sides.)(b) Prove the Pythagorean theorem:kx + yk2= kxk2+ kyk2if x⊥y.16. Find a unit vector orthogonal to the vectors x = (1, 1, 2) and y = (0, 1, 3)in R3. (Use the standard inner product.)17. Let V = C[0, 1] be the space of continuous real-valued functions definedon the interval [0, 1]. Define an inner product on C[0, 1] byhf, gi =Z10f(t)g(t) dt.(a) Verify that this does indeed define an inner product on V .(b) Evaluate kfk where f = t2− 2t + 3 ∈ V .18. Let {e1, . . . , en} be an orthonormal bas is for a complex space V , and letx ∈ V be arbitrary. Show(a) x =Pni=1eihei, xi .2(b) kxk2=Pni=1hei, xi2.19. Let V be the space of continuous complex-valued functions defined on thereal interval [−π, π], and definehf, gi =Zπ−πf∗(x)g(x) dxfor all f, g ∈ V . Show that the set of functionsfn=12π1/2einxfor n = 1, 2, . . . forms an orthonormal set.20. Let W be a subset of an inner product space V , and assume that 0 ∈ W .Prove the following:(a) 0⊥= V and V⊥= 0.(b) W ∩ W⊥= {0}.(c) W1⊂ W2implies W⊥2⊂ W⊥1.21. Consider the spac e R3with the standa rd inner product.(a) Convert the vectors u1= (1, 0, 1), u2= (1, 0, −1) and u3= (0, 3, 4) toan orthonormal basis {e1, e2, e3} of R3.(b) Write the components of an arbitrary vector x = (x1, x2, x3) ∈ R3interms of the basis {ei}.22. Let V = C[−1, 1] be the space of continuous real-valued functions definedon the interval [−1, 1]. Define an inner product on C[−1, 1] byhf, gi =Z1−1f(x)g(x) dx.Find an orthonormal basis for V generated by the functions {1, x,