Mathematics 110 Name: GSI Name: Linear Algebra Midterm Exam, April 5, 2007 Write clearly, with complete sentences, explaining your work. You will be graded on clarity, style, and brevity. If you add false statements to a correct argument, you will lose points. Be sure to put your name and your GSI’s name on every page. 1. Let V and W be finite dimensional vector spaces over a field F . (a) What is the definition of the transpose of a linear transformation T : V → W ? What is the relationship between the rank of T and the rank of its transpose? (No proof is required here.) Answer: The transpose of T is the linear transformation T t: W ∗ → V ∗ sending a functional φ ∈ W ∗ to φ ◦ T ∈ V ∗. It has the same rank as T . (b) Let T be a linear operator on V . What is the definition of a T - invariant subspace of V ? What is the definition of the T -cyclic subspace generated by an element of V ? Answer: A T -invariant subspace of V is a linear subspace W such that T w ∈ W whenever w ∈ W . The T -cyclic subspace of V generated by v is the subspace of V spanned by the set of all T nv for n a natural number. (c) Let F := R, let V be the space of polynomials with coefficients in R, and let T : V → V be the operator sending f to xf 0, where f 0 is the derivative of f. Let W be the T -cyclic subspace of V generated by x2 + 1. Compute the characteristic polynomial of T|W . Answer T (x2 + 1) = 2x2 and T (2x2) = 4x2. Thus W has a basis (x2 + 1, 2x2, and the matrix for T with respect to this basis is 0 0 . Hence the characteristic polynomial of T is x2 − 2x. 1 2 |W Mathematics 110 Name: GSI Name: 2. Let V be a finite dimensional vector space over a field F . (a) What is the definition of the dual space V ∗ of V ? If α is an or- dered basis for V , what is the definition of the dual basis of V ∗ corresponding to α? Answer: The dual space to V is the vector space of all linear transformations from V to F . If α = (v1, . , vn), then its dual is the sequence (φ1, . , φn), where φi is the unique linear transfor- mation V → F such that φi(vj) = δij. (b) Let Pn be the space of polynomials of degree at most n with coef- ficients in F . Assume that F has at least n + 1 distinct elements, c0, . , cn. For 0 ≤ i ≤ n, let φi: Pn → F denote evaluation of f at ci. Show, using properties of polynomials discussed in class and/or the book, that each φi lies in the dual space to Pn. Prove that (φ0, . , φn) is a basis for this dual space. Hint: For 0 ≤ j ≤ n, Q let fj := k6=j(x − ck) ∈ Pn. Answer: First of all φi is linear because if f, g ∈ Pn and a, b ∈ F , then φi(af + bg) = (af + bg)(ci) = af(ci) + bg(ci) = aφi(f) + bφi(g). Q Now we compute that φi(fj) = k6=j(cj − ck) = 0 iff and only if P P i 6= j. Hence if aiφi = 0, aiφi(fj) = ajφj(fj), and if it follows that aj = 0. This shows that the sequence (φ0, . φn) is linearly ∗ independent. Since we know that the dimension of Pn is n + 1, it must be a basis. (c) If F = R and in the situation of (b), c0 = −1c1 = 0, and c2 = 1, find the basis for P2 whose dual is (φ0, φ1, φ2). Answer: In fact we almost have the answer already from the hint above: f0 = x(x − 1), f1 = (x + 1)(x − 1), and f2 = (x + 1)x. These almost work, but have to be fixed by rescaling, and we find the dual basis is: x((1 − x)/2, (1 − x2), (x2 + x)/2). Mathematics 110 Name: GSI Name: 3. Let V be a vector space over a field F and let T be a linear operator on V . (a) Define the concepts of eigenvectors and eigenvalues of T . Answer: An element v ∈ V is an eigenvector of T if there exists a λ ∈ F such that T (v) = λv. An element a ∈ F is an eigenvalue of T if there is some v ∈ V with v 6= 0 such that T (v) = λv. (b) Find the eigenvalues and bases for the eigenspaces of the linear 11 −30 operator corresponding to the matrix . 4 −11 Answer: The characteristic polyonomial of T is x2 − 1, so the eigenvalues are 1 and −1. The 1-eigenspace is the nullspace of 10 −30 , which is spanned by (3, 1). The −1-eigenspace is 4 −12 12 30 the nullspace of which is spanned by (5, 2). 4 −10 (c) Find T 200, where T is the operator in the previous part. Answer: By the Cayley-Hamilton theorem, T 2 = id, so T 200 = id. 4. Let T be a linear operator on a finite dimensional vector space V over a field F . (a) What does it mean to say that T is diagonalizable? Answer: T is diagonalizable if V has a basis consisting of eigen- vectors for T . (b) Give an example of a T whose characteristic polynomial splits but which is not diagonalizable, or prove that no such example exists. 0 1 Answer: The operator corresponding to the matrix is 0 0 such an example. (c) Suppose that W is a T -invariant subspace of V . Suppose that v1 + v2 ∈ W , where v1 and v2 are eigenvectors of T corresponding to distinct eigenvalues. Prove that v1 and v2 both belong to W . Answer: Let w := v1 + v2 ∈ W . Then T (w) = λ1v1 + λ2v2 ∈ W , since W is T -invariant. We also have λ1w = λ1v1 + λ1v2 ∈ W and hence T (w) − λ1w = (λ2 − λ1)v2 ∈ W , since W is a linear subspace. Since (λ2 − λ1) 6= 0, it follows that v2 ∈ W . Since v1 = w − v2, v1 ∈ W . 4.