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CoordinateTransformation
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poyo
December 26, 2017
Education
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CoordinateTransformation
poyo
December 26, 2017
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Transcript
࠲ඪมߦྻ December 26, 2017
݁ ࠲ඪมͱجఈมͷؔ ূ໌
݁ ࠲ඪมͱجఈมͷؔ ূ໌
݁ جఈ B Ͱͷ࠲ඪ pB ͱجఈ B′ Ͱͷ࠲ඪ pB′ ͱͷؔɺB
→ B′ ͷجఈมߦྻΛ T ͱ͢Δͱ pB′ = T−1pB Ͱ͢ɻ͜ͷ T−1 ͕࠲ඪมߦྻͰ͢ɻ ಛʹඪ४جఈ E = {e1, e2, . . . , en} Λ৽͍͠جఈ B = {b1, b2, . . . , bn} ʹऔΓ͑ͨͱ͖ɺ pB = ( b1 b2 · · · bn )−1 pE ͱද͞Ε·͢ɻ
݁ ࠲ඪมͱجఈมͷؔ ূ໌
جఈมߦྻͱ جఈมߦྻ T ͱɺϕΫτϧۭؒ V ͷچجఈ B ͔Β৽جఈ B′ ͷରԠΛදͨ͠ͷͰ͢ɻ
B = {x1, x2, . . . , xn}, B′ = {y1, y2, . . . , yn} ͱ͠·͢ɻyi ͔ͨ ͕ V ͷݩͳͷͰ x1, x2, . . . , xn ͷ 1 ࣍݁߹Ͱද͞Ε·͢ɻ y1 = t11x1 + t21x2 + · · · + tn1xn y2 = t12x1 + t22x2 + · · · + tn2xn . . . yn = t1nx1 + t2nx2 + · · · + tnnxn t ͷఴ͑ࣈͷৼΓํʹҙ͠·͢ɻ
جఈมߦྻʢ͖ͭͮʣ ͦΕͧΕͷࣜ yi = ( x1 x2 · · ·
xn ) t1i t2i . . . tni ͱॻ͚ΔͷͰɺ͜ΕΛฒΔͱ ( y1 y2 · · · yn ) = ( x1 x2 · · · xn ) t11 t21 · · · tn1 t21 t22 · · · tn2 . . . . . . ... . . . tn1 tn2 · · · tnn T ͱͳΓ·͢ɻ͜ͷ T ͷٯߦྻ͕࠲ඪมߦྻʹͳΔΜͰ͢Ͷɻ͢ ͍͝ɻ
݁ ࠲ඪมͱجఈมͷؔ ূ໌
͔֬Ί Ͳ͏৴͡ΒΕͳ͍ʢͦΜͳ߹ྑ͍͜ͱ͋Δͷ͔ʁʣͷͰɺ֬ ͔ΊͯΈ·͢ɻ ͦͦ࠲ඪͱɺV ͷجఈ B = {b1, b2, .
. . , bn} ͱϕΫτϧ v ∈ V ʹରͯ͠ɺv Λ b1, b2, . . . , bn ͷ 1 ࣍݁߹Ͱදͨ͠ͱ͖ͷ ͷ͜ͱͰͨ͠ʢ࣮ͦ͏Ͱ͢ʣ ɻͭ·Γ࣍ͷΑ͏ʹରԠ͚ͮΒ Ε·͢ɻ v1 v2 . . . vn B ⇄ v1b1 + v2b2 + · · · + vnbn = ( b1 b2 · · · bn ) v1 v2 . . . vn
͔֬Ίʢ͖ͭͮʣ B Ͱͷ࠲ඪ pB ͱ B′ Ͱͷ࠲ඪ pB′ ɺͱಉ͡ϕΫτϧͳͷͰ pB
= p1 p2 . . . pn B ⇄ ( x1 x2 · · · xn ) p1 p2 . . . pn = ( y1 y2 · · · yn ) q1 q2 . . . qn ⇄ q1 q2 . . . qn B′ = pB′ ͕Γཱͪ·͢ɻ
͔֬Ίʢ͖ͭͮʣ جఈมͷࣜΛೖ͢Δͱ ( x1 x2 · · · xn )
p1 p2 . . . pn = ( x1 x2 · · · xn ) T q1 q2 . . . qn ͱͳͬͯɺߦྻܭࢉΛ͢Δͱ p1x1 + p2x2 + · · · + pnxn = k1x1 + k2x2 + · · · + knxn ͜Μͳײ͡ʹͳΓ·͢ɻ͜͜ͰɺجఈʹΑΔϕΫτϧͷදࣔҰ ҙͳͷͰ p1 = k1, p2 = k2, . . . , pn = kn ͕Γཱͪ·͢ɻ
͔֬Ίʢ͖ͭͮʣ ͭ·Γ࣍ͷ͜ͱ͕͔Γ·ͨ͠ɻ p1 p2 .
. . pn = k1 k2 . . . kn = T q1 q2 . . . qn = t11 t21 · · · tn1 t21 t22 · · · tn2 . . . . . . ... . . . tn1 tn2 · · · tnn q1 q2 . . . qn ࣮ T ਖ਼ଇͳͷͰ 1 ࠨ͔Β T−1 Λ͔͚Δͱ pB′ = T−1pB Λಘ ·͢ɻ 1B′ → B ͷجఈมߦྻΛ U ͱ͢Δͱɺ ( x1 · · · xn ) = ( y1 · · · yn ) U = ( x1 · · · xn ) TU ͱͳͬͯ TU = E Ͱ͢ɻ