MATH 218 Final, Fall 2015

MATH 218: Elementary Linear Algebra with Applications Fall 2015-2016, Final, Duration: 120 min. Exercise 1. Prove or disprove that the following transformations are linear: (a) (10 points) T1 : M2×2(R) → M2×2(R) defined by T1(A) = AAt. (b) (10 points) T2 : R2[X] → R defined by T2(P) = 2P ′(0). (c) (10 points) T3 : R1[X] → R1[X] with T3(1 + X) = X, T3(1 − X) = 1 + X and T3(2) = 2 − X. Exercise 2. Let T : R2[X] → R3[X] be the linear map defined by T(a + bX + cX2) = b − c + (a + c)X + (a + b)X2 + (a + b)X3. Consider the canonical bases C2 = {1, X, X2} and C3 = {1, X, X2, X3}. (a) (5 points) Explain briefly and with no computation why T cannot be onto. (b) (10 points) Find a basis of kerT. (c) (10 points) Find a basis of ImT. (d) (10 points) Find the matrix [T]C3 C2 of T from C2 to C3. Exercise 3. Let A =   2 0 0 −2 2 1 2 0 1  . (a) (18 points) Find the eigenvalues and eigenspaces of A. (b) (5 points) Why is A diagonalizable? (c) (10 points) Find P invertible and D diagonal such that A = PDP −1 (do not compute P −1). Exercise 4. (a) Let A = � 2 −1 1 0 � . i. (5 points) Show that 1 is an eigenvalue of multiplicity two. ii. (3 points) Without determining the eigenspace V1 nor its dimension, explain why A is not diago- nalisable (hint: if A was diagonalisable then there would be P, D such that...). (b) (5 points) Let A be a 2 × 2 matrix. Assume that A has two distinct eigenvalues 1 and −1. Prove that A2 = I. (c) (4 points) Find a 3 × 3 matrix A with only two eigenvalues 1 and −1 and such that that A2 ̸= I. (d) (5 points) Determine one eigenvalue of A =       −1 1 1 0 2 3 2 −3 1 0 2 5 −2 0 3 −1 0 1 1 −2 2 2 −2 −1 1       . Exercise 5. Let B =      1 0 1   ,   2 −1 0   ,   0 2 1     . (a) (10 points) Show that B is a basis of R3. (b) (15 points) Using the Gram-Schmidt process to transform B into an orthonormal basis. 2 Exercise 6. Consider the subspace W = span      1 −1 0   ,   0 1 1     . (a) (10 points) Determine a basis of W ⊥. (b) (5 points) Let v =   0 1 1  . Find projW (v). (c) (5 points) Find explicitly w ∈ W and w′ ∈ W ⊥ such that v = w + w′. Exercise 7. Assume that R3 is endowed with the dot product and let W = span{w} be a subspace of R3 of dimension one. Let projW : R3 → R3 be the orthogonal projection on W. (a) Part A. The goal is to show that projW is diagonalisable. i. (3 points) Prove that if u ∈ W then projW (u) = u (hint: if u ∈ W = span{w} then u can be written u = cw where c is a scalar). ii. (4 points) Deduce that 1 is an eigenvalue and show that the eigenspace V1 associated to 1 is equal to W. iii. (2 points) Deduce also from i. that Im projW = W. iv. (4 points) What is the dimension of W ⊥? v. (3 points) Prove that Ker projW = W ⊥. vi. (3 points) Use v. to determine another eigenvalue of projW and the corresponding eigenspace. vii. (3 points) Why is projW diagonalisable? (b) Part B. The goal is to find a basis B in which [projW ]B is diagonal. Assume that {v1, v2} is an orthog- onal basis of W ⊥. i. (4 points) Prove that B = {w, v1, v2} is an orthogonal basis of R3 (note: recall that we have W = span{w}). ii. (4 points) Determine the matrix representation [projW ]B of projW in the basis B (c) Part C. (5 points) Consider the particular case W = span      −1 1 1     . Find a basis B such that [projW ]B is diagonal (hint: this part is a direct application of Part B; e.g. the first step is to find an orthogonal basis of W ⊥,...).