MATHEMATICS 2B (MATH248) 2020 TUTORIAL 9 Text Problems Page 37, nos. 24(b,c), 25, 26, 27 Page 76, nos. 28, 29, 31, 32, 33(i), 34, 36, 37, 38, 39, 41 Additional Problems 0 1 1 1. A 3 × 3 matrix A has eigenspaces E1 = 0 , 1 and E−2 = 2 . Use this 0 1 1 10 information to calculate A . 1 1023 −1023 (Ans: 0 2047 −2046 ) 0 1023 −1022 2. Use matrix methods to solve the following systems of differential equations (all functions are functions of t): (a) x′ = x + 3y y ′ = 2x + 2y subject to x(0) = 0, y(0) = 5. (Ans: x(t) = −3e−t + 3e4t , y(t) = 2e−t + 3e4t ) (b) x′ = x + 3z y ′ = x − 2y + z z ′ = 3x + z (Ans: x(t) = 3e4t −e−2t , y(t) = e4t +2e−2t , subject to x(0) = 2, y(0) = 3, z(0) = 4. z = 3e4t + e−2t ) 3. A scientist places two strains of bacteria, X and Y , in a petri dish. Initially there are 400 of X and 500 of Y . If x = x(t) and y = y(t) are the numbers of the strains at time t days, the growth rates of the two populations are given by the system x′ = 1.2x − 0.2y y ′ = −0.2x + 1.5y . Determine what happens to these two populations by solving this system (use matrix methods). (Ans: x(t) = −120e8t/5 + 520e11t/10 , y(t) = 240e8t/5 + 260e11t/10 ; so strain X dies out after 2 ln(13/3) days and strain Y continues to grow) 2 3 4. W is a subspace of R defined by W = span −3 . 1 −1 3 (Ans: 2 , 0 ) (a) Find a basis for W ⊥ . 0 (b) Describe W ⊥ geometrically. 2 (Ans: the plane 2x − 3y + z = 0) 5. Obtain theorthogonal complement of subspace W ofR4 defined by −1 1 2 1 0 1 0 −2 W = span ) , , . (Ans: span 0 −2 3 0 1 1 4 1 6. Consider the vectors v1 = 1 4 5 6 9 , v2 = −1 0 −1 −2 −1 and v3 = 2 3 5 7 8 . 5 (a) Obtain the orthogonal ofthe complement subspace W = span (v1 , v2 , v3 ) of R . −1 −2 −1 −1 −1 −2 (Ans: span 1 , 0 , 0 ) 0 1 0 0 0 1 (b) What is the dimension of W ? Explain. 7. Subspace W in R4 is described by the equation x + y + z − w = 0. −1 −1 1 1 0 0 , , ) (Ans: 0 1 0 (a) Find a basis for W . 0 (c) Express b = 4 4 5 13 + 8. Let M = −2 −2 −2 2 2 2 3 15 1 1 1 ) (Ans: 1 (b) Obtain a basis for W ⊥ . 0 −1 as the sum of a vector in W and a vector in W ⊥ . ) 2 −1 1 0 2 −1 1 0 −1 2 −2 1 0 −3 0 1 1 1 −1 1 . (Ans: (a) Obtain a basis for the orthogonal complement of the column −1 1 subspace spanned by the columns of M . (Ans: span 0 0 0 space , i.e. the of M −1 0 , −1 0 ) 1 (b) Hence deduce whether the columns of M form a linearly independent set of vectors.
