tut_tt1_review - Math 235 Tutorial Term Test 1 Review 1...

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Math 235 Tutorial: Term Test 1 Review 1: State the definition of: a) One-to-one b) Onto c) An orthogonal matrix (what are 2 other equivalent definitions?) d) An inner product 2: Let B = 1 / 2 1 / 2 1 / 2 1 / 2 , 1 / 2 - 1 / 2 1 / 2 - 1 / 2 , - 1 2 0 1 2 0 = { ~v 1 ,~v 2 ,~v 3 } be vectors in M (2 , 2) under the inner product h A, B i = tr( A T B ). (a) Show that { ~v 1 ,~v 2 ,~v 3 } is an orthogonal basis for span B . (b) Is { ~v 1 ,~v w ,~v 3 } an orthonormal basis for span B ? (c) Find the coordinates of 4 1 0 1 with respect to B . 3: Consider L : P 1 M (2 , 2) defined by L ( a 1 x + a 2 ) = a 1 a 2 - a 2 a 1 . a) Find a basis for the range and nullspace of L and verify the Rank-Nullity theorem. b) Find the matrix of L with respect to the basis B = { x + 1 , x - 1 } of P 1 and C = 1 0 0 1 , 0 1 0 0 , 0 0 1 0 , 0 0 1 1 of M (2 , 2). c) Use your answer in b ) to find the
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