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# tut_tt1_review_ans - Math 235 Tutorial Term Test 1 Review W...

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Unformatted text preview: Math 235 Tutorial: Term Test 1 Review W P 0\¢\‘ m5 do 3 1 ’13 f 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 1/2 1/2] [1/2 4/2] [‘4 0]} 1 1 1 . 2:LetB= , ,ﬁ =v,v,'u b t M IR {[1/2 1/2 1/2 —1/2 % 0 { 1 2 3} GM (”8m 2x2( ) under the inner product (A, B) = tr(BTA). (a) Show that {171,272,173} is an orthogonal basis for span [3. (b) Is {171,172,173} an orthonormal basis for span 8? 41 (c) Find the coordinates of [0 1 with respect to B. ~a2 a1 3: Consider L : P1 —> MQXZOR) deﬁned by L(a1\$ + a2) : |: a1 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 8 = {x + 1,3: — 1} of P1 and His 2112 3112 3112 211mm» c) Use your answer in b) to ﬁnd the C—coordinates of L(f) where [\$13 = [g] . 4: Let A be an m X n matrix such that the dimension of the nullspace of A is 7". Determine, with proof, the dimension of the nullspace of AT. 5: Let T be a linear operator on an inner product space V, and suppose that (if?) = (T(i’), T(g)’)) for all a? and 37 in V. Prove that T is an isomorphism. m H sax M‘ «UL M WHAT)? gmcgwm z. JIM (mu. UM} : MAM (A) M a: (MM) W M «T M- WHAT).q‘mmwqu) 7 :M;{n_r‘)' \\\ ’ Lav T? qu i For I am 3:73; PF‘ W 55X 40 's \‘i ...
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tut_tt1_review_ans - Math 235 Tutorial Term Test 1 Review W...

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