fylladio2-2005 - Γραµµική Άλγεβρα 2ο...

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Unformatted text preview: Γραµµική Άλγεβρα 2ο Φυλλάδιο Ασκήσεων (ΟΡΙΖΟΥΣΕΣ) ΣΗΜΜΥ 2005-06 xabx axxb 1. Να αποδειχθεί ότι = (b − a) 2 (a + b + 2 x)(a + b − 2 x) . bxxa xbax xabc axbc =0. 2. Να λυθεί η εξίσωση abxc abcx 3. Να υπολογιστεί η n × n ορίζουσα x+k x x … x+k … x 4. Να υπολογιστεί η n × n ορίζουσα x x2 . … x+k 1 + x1 x2 … x1 1 + x2 … x1 ⎡0 ⎢0 ⎢ ⎢0 5. Έστω ο n × n πίνακας C = ⎢ ⎢ ⎢0 ⎢ ⎢− a0 ⎣ x x xn xn . … 1 + xn 1 0 0 0 1 0 0 0 1 0 0 0 − a1 − a2 − a3 0⎤ … 0⎥ ⎥ … 0⎥ ⎥ . Να αποδειχθεί ⎥ 1⎥ ⎥ … − a n −1 ⎥ ⎦ … ότι | xI n − C | = x n + a n −1 x n −1 + a n − 2 x n − 2 + … + a1 x + a 0 . 6. Έστω A ∈ Π n και X , Y ∈ Π n×1 . Να αποδειχθεί ότι για κάθε µιγαδικό ξ , ισχύει ⎡A det ⎢ T ⎣Y X⎤ T ⎥ = ξ ⋅ det A − Y (adjA) X . ξ⎦ Σ. Καρανάσιος Endeiktikèc LÔseic FulladÐou 02 1. x a b x a x x b b x x a x b a x = a + b + 2x a + b + 2x b + 2x + a 2x + b + a a x x b b x x a x−a x−b b−x (a + b + 2x) x − a x − b a − x b−a a−b 0 x b a x 1 a b x 0 x−a x−b b−x = (a + b + 2x) 0 x−a x−b a−x 0 b−a a−b 0 x−a x−b b−x = (a + b + 2x)(b − a) x − a x − b a − x 1 −1 0 0 0 b−a (a + b + 2x)(b − a) x − a x − b a − x 1 −1 0 = (a + b + 2x)(b − a)2 x−a x−b 1 −1 = = = (a + b + 2x)(b − a)2 (a + b − 2x). 2. x a a a a x b b b b x c c c c x = x+a+b+c x+a+b+c x+a+b+c x+a+b+c a x b b x−a 0 0 0 (x+a+b+c) a − b x − b b−a c−b c−x b b x c c c c x 1 a b c 0 x−a 0 0 = (x + a + b + c) 0 a−b x−b 0 0 b−a c−b c−x = (x+a+b+c)(x−a) x−b 0 c−b c−x = = (x+a+b+c)(x−a)(x−b)(c−x) 3. x+k x x x+k . . . . . . x x ··· ··· ··· ··· x x . . . x+k 1 x ··· 0 k ··· = (k +nx) . . . . ··· .. 0 0 ··· x 0 . . . k k 0 ··· 0 k ··· = (k +nx) . . . . ··· .. 0 0 ··· 0 0 . . . = (k +nx)k n k 4. Ergazìmenoi akrib¸c ìpwc stic dÔo prohgoÔmenec ask seic, brÐskoume ìti h orÐzousa isoÔtai me 1 + x1 + x2 + · · · + xn 5. |xIn −C | = x 0 0 . . . 0 a0 −1 0 0 x −1 0 0 x −1 . . . . . . . . . 0 0 0 a1 a2 a3 ··· ··· ··· . . . 0 0 0 . . . ··· ··· −1 x + an = (pol/me thn 2h st lh epÐ x, thn 3h epÐ x2 k.o.k. thn teleutaÐa st lh epÐ xn−1 kai tic prosjètoume sthn 1h st lh, opìte prokÔptei h orÐzousa) = 0 0 0 . . . −1 0 0 x −1 0 0 x −1 . . . . . . . . . 0 0 0 0 p(x) a1 a2 a3 ··· ··· ··· . . . 0 0 0 . . . ··· ··· −1 x + an , ìpou p(x) = a0 + a1 x + · · · + an−1 xn−1 + xn . AnaptÔssontac thn orÐzousa wc proc ta stoiqeÐa thc 1hc gramm c paÐrnoume: |xIn − C | = (−1)n+1 p(x)(−1)n−1 = p(x) = a0 + a1 x + · · · + an−1 xn−1 + xn . 6. D= A Y X ξ = a11 . . . ··· . . . an1 · · · y1 · · · a1n . . . x1 . . . ann xn yn ξ . AnaptÔssontac thn teleutaÐa orÐzousa wc proc ta stoiqeÐa thc teleutaÐac gramm c, qrhsimopoi¸ntac ton tÔpo (6.7) tou biblÐou, (sel. 168), èqoume: D = y1 D1 + · · · + yn Dn + ξdetA, (1) ìpou Di to algebrikì sumpl rwma tou stoiqeÐou yi . An t¸ra anaptÔxoume kˆje mia apì tic orÐzousec Di wc proc ta stoiqeÐa thc teleutaÐac st lhc paÐrnoume: n y1 D1 + · · · + yn Dn = y1 ( n xi Ai1 ) + y2 ( i=1 [y1 · · · yn ] A11 A12 . . . A1n A21 · · · A22 · · · . . . . . . A2n · · · Apì (1),(2) prokÔptei to zhtoÔmeno. n xi Ai2 ) + · · · + yn ( i=1 An1 An2 . . . Ann x1 x2 . . . xn xi Ain ) = i=1 = Y (adjA)X. (2) ...
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This note was uploaded on 10/02/2009 for the course G 001 taught by Professor Shmmygr during the Spring '07 term at National Technical University of Athens, Athens.

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