Indian Institute of Technology Kharagpur Department of Mathematics MA11004 - Linear Algebra, Numerical and Complex Analysis Problem Sheet - 2 Spring 2022 1. Determine which of the following form a basis of the respective vector spaces: (a) {4t2 − 2t + 3, 6t2 − t + 4, 8t2 − 8t + 7} of P2 (R). (b) Let V be a real vector space with {α, β, γ} as a basis. Check whether {α + β + γ, β + γ, γ} is also a basis of V . (c) 1 0 0 0 1 1 1 1 1 1 , , , 0 0 1 0 1 1 for V , where V is the vector space of all 2 × 2 real matrices. 2. Determine a basis and the dimension of the following subspaces: (a) The subspace V of all 2 × 2 real symmetric matrices. (b) U = {(x, y, z, w) ∈ R4 : x + 2y − z = 0, 2x + y + w = 0} of R4 . Z 1 (c) Let U = {p ∈ P4 (R) : p(t)dt = 0}. −1 3. Let S = {(x, y, z, w) ∈ R4 : x+y+z+w = 0} and T = {(x, y, z, w) ∈ R4 : 2x+y−z+w = 0} be two subspaces of R4 . Find the dimension of S ∩ T . 4. Examine whether T is a linear transformation or not (i) T : R2 → R2 defined by T (x, y) = (x + y, x − y), (x, y) ∈ R2 . (ii) T : R2 → R3 defined by T (x, y) = (x + 2y, 2x + y, x + y), (x, y) ∈ R2 . (iii) T : R3 → R3 defined by T (x, y, z) = (yz, zx, xy), (x, y, z) ∈ R3 . 5. Consider the vector space C over the field C. Give an example of a function φ : C → C such that φ(w + z) = φ(w) + φ(z) ∀ w, z ∈ C, but φ is not a linear transformation . 6. Find the null space and range space of the following linear transformations. Also find their respective dimensions and verify the rank-nullity theorem. Z x (a) T : P2 (R) → P3 (R) defined by T (f (x)) = 2f 0 (x) + 3f (t)dt. 0 3 2 (b) T : R → R , defined by T (x, y, z) = ( x−y−z , z2 ). 2 (c) T : M2×2 (R) → M2×2 (R) defined by T (A) = A − AT , ∀A ∈ M2×2 (R). 2 7. (a) Determine the linear transformation T : R2 → R3 such that T (1, 1) = (1, 0, 2), T (2, 3) = (1, −1, 4). (b) Determine the linear transformation T : R3 → R2 which maps the basis vectors {(1, 0, 0), (0, 1, 0), (0, 0, 1)} of R3 to the vectors {(1, 1), (2, 3), (3, 2)} respectively. 1 (i) Find T (1, 1, 0), T (6, 0, −1), (ii) Find N (T ) and R(T ), (iii) Prove that T is not one-to-one but onto. 8. Find the matrix of the linear transformations with respect to the given ordered bases: (a) D : P4 (R) → P4 (R) defined by D(p(x)) = 3 basis {1, x, x2 , x3 , x4 } for both P4 (R). (b) T : P3 (R) → M2×2 (R) by d3 (p(x)), with respect to the ordered dx3 00 2f (0) f (3) T (f (x)) = 0 f 0 (2) 1 0 0 1 0 0 0 0 2 3 with respect to the ordered basis {1, x, x , x } and , , , . 0 0 0 0 1 0 0 1 9. Determine the linear mapping T : R3 → R4 that maps the basis vector (0,1,1), (1,0,1), (1,1,0) of R3 to the vectors (0,1,1,1), (1,0,1,1), (1,1,0,1), respectively. Find null space N (T ) and range space R(T ) and also verify that dim N (T ) + dim R(T ) = 3. 10. Find the rank of the following matrices 1 3 1 −5 1 −2 1 −2 1 −5 (ii) (i) −1 1 5 −7 2 1 −1 2 −2 (iv) 2 (iii) 4 −3 4 3 −2 4 −4 [Answers. (i) 3 (ii) 2 (iii) 2 −2 3 4 4 −1 −3 2 7 6 2 −1 3 4 1 −2 6 3 −7 (iv) 2 ] 11. Use Gauss elimination method to conclude about the (in)consistency of the system x + y − 3z = 4 2x + y − z = 2 3x + 2y − 4z = 7 [Hint. rank A = 2 and rank of the augmented matrix is 3, so inconsistent] 12. Use Gauss elimination method to show that the following system has a unique solution. Find also the solution. 2x − 2y = −6 x−y+z = 1 3y − 2z = − 5 13. Show that the system x + 2y + 4z + w = 4 2x − z − 3w = 4 x − 2y − z = 0 3x + y − z − 5w = 5 2 is inconsistent. [Hint. rank A = 3 and rank of the augmented matrix is 4] 14. Determine the condition for which the system x+y+z =b 2x + y + 3z = b + 1 5x + 2y + az = b2 admits of (i) only one solution, (ii) no solution, (iii) infinitely many solutions. ********** 3