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tut_tt1_review

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

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Unformatted text preview: 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 − √1 0 1/2 1/2 1/2 −1/2 , , √2 1 1/2 1/2 1/2 −1/2 0 2 T under the inner product A, B = tr(B A). 2: Let B = = {v1 , v2 , v3 } be vectors in M (2, 2) (a) Show that {v1 , v2 , v3 } is an orthogonal basis for span B . (b) Is {v1 , v2 , v3 } an orthonormal basis for span B ? (c) Find the coordinates of 41 with respect to B . 01 a1 a2 . − a2 a1 3: Consider L : P1 → M (2, 2) deﬁned by L(a1 x + a2 ) = 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 P1 and 01 10 00 00 , , of M (2, 2). , C= 00 10 11 01 c) Use your answer in b) to ﬁnd the C -coordinates of L(x) where [x]B = 3 . 2 4: Let B = 1 −2 , −3 4 such that P [x]B = [x]C . and C = −7 −5 , 9 7 be bases for R2 . Find the matrix P 5: Let A be an m × n matrix such that the dimension of the nullspace of A is r. Determine, with proof, the dimension of the nullspace of AT . 6: Let T be a linear operator on an inner product space V , and suppose that x, y = T (x), T (y ) for all x and y in V . Prove that T is an isomorphism. ...
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