Linear Algebra I Problem Set 3: Vector Spaces Dr Nicholas Sedlmayr Friday February 5th 2016 Due: In class, February 12th 2016 1. Prove proposition 2.9 from the lectures which states the following: Suppose A = {a1 , a2 , . . . an } V is linearly independent, where V is a vector space over F . Suppose also that v V and there are scalars 1 , . . . n and 1 , . . . n such that v = 1 a1 + 2 a2 + . . . n an and v = 1 a1 + 2 a2 + . . . n an then 1 = 1 , 2 = 2 ,. . . n = n . (4) (Tip: consider the denition of linear independence.) 2. Prove that the set of Pauli matrices and the 2 2 identity matrix, A = {I2 , x , y , z }, is a basis of the vector space V , the set of all 22 matrices over C, with addition and scalar multiplication dened in the usual way. (4) The Pauli matrices are x = 01 10 , y = 0 i i 0 , z = 1 0 0 1 . 3. Find a basis for the following vector spaces: (a) The set of all 2 2 matrices over R. (2) (b) C4 , i.e. the set of 4 1 column vectors with complex entries. (2) 4. Which of the following are bases over R3 ? Give reasons! (4) (a) (b) (c) (d) A = {(1, 1, 0)T , (0, 1, 0)T , (1, 0, 1)T , (0, 0, 1)T }. B = {(1, 1, 0)T , (0, 1, 0)T , (1, 0, 1)T }. C = {(1, 0, 0)T , (0, i, 0)T , (1, 0, i)T }. D = {(1, 1, 1)T , (2, 2, 1)T , (1, 1, 0)T , }. 5. Find bases over the following subspaces of R3 . (4) (a) A = {(x, y, z)T : 2x + y z = 0}. (b) B = {(x, y, z)T : x + y 2z = 0, x y = 0, }. Total available marks: 20 1