(1)TRENTO, A.A. 2019/20 MATHEMATICS FOR DATA SCIENCE/BIOSTATISTICS EXERCISE SHEET # 10 Important! In solving the exercises • explain what you are doing, • explain why you are doing what you are doing, and • spell out all intermediate steps. Notice! For technical reasons, sometimes in the following exercises we write something like “e1+ ( −1)e2”, which just means e1− e2. Exercise 10.1. Let M2×2 be the vector space of 2 × 2 real matrices. We recall that dim(V ) = 4. • Prove that the matrices A = [ 0 1 2 0 ] , B = [ 1 2 3 0 ] are linearly independent. • Extend A, B to a basis of M2×2. Exercise 10.2. Let M2×3 be the vector space of 2 × 3 real matrices. We recall that dim(V ) = 6. • Prove that the matrices A = [ 0 1 2 2 0 0 ] , B = [ 1 2 4 3 0 1 ] , C = [ 1 0 1 0 1 1 ] are linearly independent. • Extend A, B, C to a basis of M2×3. Exercise 10.3. Let M2×2 be the vector space of 2 × 2 real matrices. Let V = span {A, B, C, D}, where A = [ 0 1 2 0 ] , B = [ 1 2 3 0 ] , C = [ 1 2 3 4 ] , D = [ 1 2 3 1 ] . Extract from A, B, C, D a basis of V . Is V = M2×2? Exercise 10.4. Let V be a vector space of dimension 2, and let e1, e2 be a base of V . Consider the linear function f : V → V given by { f (e1) = e1+ e2 f (e2) = ( −1)e1+ 2e2 Write the matrix A of f with respect to the base e1, e2. (2)2 EXERCISE SHEET # 10 Consider the vectors { g1 = e1+ ( −1)e2, g2 = 2e1+ e2. (1) Show that g1, g2 are a base of V , (2) write e1, e2 as linear combinations of g1, g2, and (3) write the matrix B of f with respect to the base g1, g2. Exercise 10.5. Let V be a vector space of dimension 2, and let e1, e2 be a base of V . Consider the linear function f : V → V given by { f (e1) = 5e1+ 2e2 f (e2) = ( −12)e1+ ( −5)e2 Write the matrix A of f with respect to the base e1, e2. Consider the vectors { g1 = 3e1+ e2, g2 = 2e1+ e2. (1) Show that g1, g2 are a base of V , (2) write e1, e2 as linear combinations of g1, g2, and (3) write the matrix B of f with respect to the base g1, g2. Exercise 10.6. Let V be a vector space of dimension 2, and let e1, e2 be a base of V . Consider the linear function f : V → V given by { f (e1) = ( −4)e1+ 10e2 f (e2) = ( −3)e1+ 7e2 Write the matrix A of f with respect to the base e1, e2. Consider the vectors { g1 = 3e1+ ( −5)e2, g2 = ( −1)e1+ 2e2. (1) Show that g1, g2 are a base of V , (2) write e1, e2 as linear combinations of g1, g2, and (3) write the matrix B of f with respect to the base g1, g2. Exercise 10.7. Use the rules of Laplace and Sarrus to compute some of the deter- minants of sheet #8. Exercise 10.8. Let V be a vector space of dimension 3, and let e1, e2, e3 be a base of V . Consider the linear function f : V → V given by f (e1) = e1 + e2 f (e2) = e2 + e3 f (e3) = e1 − e3 (3)EXERCISE SHEET # 10 3 Write the matrix A of f with respect to the base e1, e2, e3. Consider the vectors g1 = e1, g2 = e1 − e2 − e3, g3 = e3. (1) Show that g1, g2, g3 are a base of V , (2) write e1, e2, e3 as linear combinations of g1, g2, g3, and (3) write the matrix B of f with respect to the base g1, g2, g3. Read more
