/ 2 n n n n x 1,..., x n 1 n 2 n R n n ndimensional Euclidean space R n vector point R n set space R n R n x = x 1 x n y = y 1 y n distance dx,

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1 1 1.1 R n xyz xyz 3 x, y, z R 3 := x y : x, y, z R z 1 3. n n x 1,..., x n x 1. x n x 1 x n 1

2 / 2 n n n n x 1,..., x n 1 n 2 n R n n ndimensional Euclidean space R n vector point R n set space R n R n x = x 1 x n y = y 1 y n distance dx, y := x 1 y x n y n 2 Euclidean distance n = x 1. x n = t x 1 x n t n 1 transverse 1 n t x 1 x n = x 1. x n

3 / 3 R n R n equip R n x x y dx, y < dx, y x x x x R n. R 2 ABCD A B B C AB = BC R A B D C AB = DC R 2. AB + BC = 2AB R AB + DC = 2 AB R 2 2 3

4 / 4. R n R n R n a + b = b + a 3 R n a b c a + b = c R n. V R vector space V1 a, b V a + b V unique V2 α R, a V αa V V3 a, b, c V, α, β R 3 1km 1km 1km 1km

5 / 5 1. a + b + c = a + b + c. 2. a + b = b + a V 0 + a = a + 0 = a 4. a V a V a + a = a + a = 0 5. α + βa = αa + βa. 6. αa + b = αa + αb. 7. αβa = αβa 8. 1 a = a V vector R C C R n V1-V3 R V1-V3 1 R n 2 M n R := {A = A : A n } 3 R := {a = x 1, x 2,... : x j R} 4 F := {a = x 1, x 2,... R : x j+2 = x j+1 + x j } 5 C 0 R := {f = fx : fx R R } 6 C 1 R := { f = fx C 0 R : f x C 0 R } 7 Poly := { f = fx C 1 R : fx } 8 Poly d := {f = fx Poly : fx d } C 1 R C 0 R vector subspace

6 / R n O P O P 1. O u 1, u 2 2. u 1, u 2 2 l 1, l 2

7 / 7 3. l 1, l 2 u 1, u 2 4. X l 1, l 2 OX = x 1 u 1 + x 2 u 2 x 1, x 2 R 2 5. x 1, x 2 R 2 x 1 u 1 + x 2 u 2 X = OX R 2 R 2 u 1, u 2 P OP = p 1 u 1 + p 2 u 2 p 1, p 2 p 1, p 2 OP : O P. u 1, u 2 basis

8 / 8 V {u 1,..., u n } V V basis a V x 1,..., x n R n a = x 1 u x n u n = u 1 u n. x 1 x n 1.1 R n x 1,..., x n x u 1,..., u n coordinate value V n n-dimensional 1.1 u 1 u n x 1. x n V n V R n V OP Aretha u 1, u 2 OP = a 1 u 1 + a 2 u 2 = u 1 u 2 a 1, a 2 u 1, u 2 P a1 a 2 Otis v 1, v 2 OP = b 1 v 1 + b 2 v 2 = v 1 v 2 a1 a 2 b1 b 2 4 R n

9 / 9 b 1, b 2 OP = u 1 u 2 a1 a 2 = v 1 v 2 {u 1, u 2 } {v 1, v 2 } a1 a 2 b1 b 2 mm, cm, m, km L L = 2cm = 20mm cm mm 2 20 cm 1 10 = mm 2 10 = OP OP = u 1 u 2 a1. a 2 = v 1 v 2 b1 b 2 b1 b 2 = 1cm= 10mm 2cm 20mm m 2 65m Otis v 1, v 2 Aretha u 1, u 2 v 1 = u 1 u 2 q11 q 21, v 2 = u 1 u 2 q12 q ij v 1 v 2 = u 1 u 2 q11 q 12 q 21 q 22 Q Q 0 v 1 v 2 q 22

10 / 10 Q 1 v 1 v 2 = u 1 u 2 Q u 1 u 2 = v 1 v 2 Q 1 10mm=cm OP OP = u 1 u 2 a1 a 2 = v 1 v 2 b1 b 2 u 1 u 2 a1 a 2 = v 1 v 2 Q 1 Q b1 b 2 = u 1 u 2 Q b1 b 2 u 1 u 2 a1 b1 b1 = Q = Q 1 a1 a 2 b 2 Aretha u 1 u 2 Otis v 1 v 2 Q Q 1 a 1, a 2, b 1, b 2 Q Otis Aretha b 2 a V R / V {u 1,..., u n } {v 1,..., v n } 5 1 k n v k {u 1,..., u n } q 1k v k = u 1 u n. 5 n q nk

11 / 11 q 11 q 1n v 1 v n = u 1 u n..... q n1 q nn Q := q ij 1 i,j n a V a 1 1 a = u 1 u n. = v 1 v n b. a n v 1 v n = u 1 u n Q, 1 b. b n b n = Q 1 a 1. a n U V f : U V U V U fu 6 f : U V linear map a, b U, α R L1 fa + b = fa + fb L2 fαa = αfa L1 6

12 / 12 U a b a + b f V fa fb fa + fa 7 U = R m, V = R n n m A f : R m R n, x Ax n = m m = n = 1 A f : R R; fx = Ax fαx = αfx α fx + y = fx + fy. f : R R, fx = x f : U V isomorphism U V V U f V 3 {u 1, u 2, u 3 } a, b V a = a 1 u 1 + a 2 u 2 + a 3 u 3 V a 1, a 2, a 3 R 3 ϕ = ϕ {u1,u 2,u 3 } : V R 3 ϕ : a a 1, a 2, a 3 ϕ a 1, a 2, a 3 R 3 ϕa 1 u 1 + a 2 u 2 + a 3 u 3 = a 1, a 2, a 3 ϕ a = a 1 u 1 + a 2 u 2 + a 3 u 3 V ϕa = ϕa ϕ a i = a i a = a i = 1, 2, 3 7 L1 L2

13 / 13 b = b 1 u 1 + b 2 u 2 + b 3 u 3 V α R a 1 + b 1 a 1 b 1 ϕa + b = a 2 + b 2 = a 2 + b 2 = ϕa + ϕb a 3 + b 3 a 3 b 3 αa 1 a 1 ϕαa = αa 2 = α a 2 = αϕa αa 3 a 3 8 ϕ : V R 3 V R 3 V ϕ. ϕ : V U V U cm V U ϕ V U 1.6 n = 2 3 R n a = [a 1 a n ], b = [b 1 b n ] inner product a b := a 1 b 1 + a 2 b 2 R a b := a 1 b 1 + a 2 b 2 + a 3 b 3 R a b = b a a b a b 8 ϕ

14 / R n n = 2 x = x, y R 2 xy x x x y x 1 f 1 x := x R 1 1 f 2 x := x R 2 f 1 x = 3.8 f 2 x = x = x + y = y 1 x = x 2y = y x = 0.8, y = 3.0 R 2 R R n n

15 / f 1 : R 2 R, f 1 x = 1 x R x x = x, y k R f 1 x = x + y = k 1 f 1 R : f 1 x = x + y x k 0.9 f 1 x f 1 x x x 10 f 1 x = f 1 x f 2 f 2 x f 2 x 10

16 / 16 a = a, b f a : R 2 R, f a x = a x R a = 0 {ax + by = k} k R a = a, b f a f a : R 2 R f a αx+βy = αf a x+βf a y f a 0 = 0 f a x = f a x x x a. a f a a f a = f a a = a R n f : R n R R n linear functional R n f a R n g : R n R a R n g = f a. n = 7 {e 1,..., e 7 } R 7 x R 7 x = x 1 e x 7 e 7 gx = x 1 ge x 7 ge 7 = ge 1. ge 7 a = ge 1,, ge 7 gx = a x = f a x x a x 1. x 7 a R n f a g R n a R n R n

17 / a : ge 1 x 1 gx = x 1 ge x n ge n =.. ge n gx = x 1 ge x n ge n = ge 1 ge n 1 n R n g ge 1 ge n R n a R n a = a 1 a n R n x R n f a x = ax a, x R n a x = t ax a x n x 1. x n a, b R n α, β R αa + βb x = αa x + βb x x R n R n f αa+βb x = αf a x + βf b x x R n 11

18 / 18 R n f, g R n α R f + g R n f + gx := fx + gx x R n αf R n αfx := αfx x R n R n R n R n dual space R n x R n 0 R f, g : R 2 R e 1 = 1, 0 fe 1 ge 1 f g e 1 = 1, 0 f R n fe 1 R f : R 2 R f e 1 = 1, 0 e 1 e 2 = 0, 1 fe 1 = 3.1, fe 2 = 4.8 x = x, y R 2 fx = fxe 1 + ye 2 = xfe 1 + yfe 2 12 R n R n

19 / 19 fx = 3.1x + 4.8y f e 1 e 2 f x R 2 {0}. {u 1, u 2 } R 2 fu 1, fu 2 f. f R n x R n x x R n f R n f R n R n dual space V. f : V R f V linear functional x, y V α R LF1 fx + y = fx + fy R LF2 fαx = αfx R V LF1 LF2. V = R f : V R x = a 1, a 2,... V fx := a 7 R

20 / 20 f LF1 LF2 7 V 7. V = Poly 2 f : V R x = xt V fx := x0 R f LF1 LF2 2 t = 0. V = Poly 2 f : V R x = xt V fx := 1 0 txtdt R f 2 x = xt 1.2. V V dual space V f, g V, α R DS1 f + g V x V fx + gx R DS2 αf V x V αfx R V. V V x y f V fx fy x y V V f g x V fx gx f g V V

21 / V {u 1,..., u n } V V {f 1,... f n } V f i u j = δ ij := { 1 i = j 0 i j dim V = dim V. {f 1,..., f n } {u 1,..., u n } dual basis. n = 6 x V x = x 1 u 1 + x 6 u 6 = u 1 u 6. x 1 x 1,..., x 6 x f i : V R i = 1,..., 6 x 1 x 6 f i : x = u 1 u 6. x i x V {u 1,..., u 6 } i f i u j = δ ij {f 1,..., f 6 } V g V α j = gu j j = 1,..., 6 gx = gx 1 u x 6 u 6 = x 1 gu x 6 gu 6 = x 1 α x 6 α 6. α 1 f α 6 f 6 x = α 1 f 1 x + + α 6 f 6 x = α 1 x α 6 x 6. x V g = α 1 f α 6 f 6 V g {f 1,..., f 6 } α 1,..., α 6 R 6 g = α 1 f α 6 f 6 V u j j = 1,..., 6 gu j = α j = α j {f 1,..., f 6 } {f 1,..., f 6 } V x 6

22 / R V, : V V R a, b, c V α R IN1 a, b = b, a IN2 a + b, c = a, c + b, c IN3 α a, b = αa, b = a, αb IN4 a, a 0 a = 0 inner product a, a a a a, b = 0 a b R n n R n a = a 1,..., a n, b = b 1,..., b n R n b n 1 a b := a 1 a n b. = a 1b a n b n R a b canonical inner product a b a, b R n P n x P x, P : R n R n R x, y P := P x, P y R n R x P x IN1 IN4 R n I = [ 1, 1] R C 0 I f = ft, g = gt C 0 I f, g := ftgtdt R I 13 2 f : R 2 R, fx, y = xy 2 f : V V R

23 / 23 V, : V V R a V f a : x a, x R V V V g V a V g = f a. V V. Hint: - V {u 1,..., u n } {f 1,..., f n } f i x = u i, x g = i α if i a = i α iu i g = f a

x V x x V x, x V x = x + = x +(x+x )=(x +x)+x = +x = x x = x x = x =x =(+)x =x +x = x +x x = x ( )x = x =x =(+( ))x =x +( )x = x +( )x ( )x = x x x R

x V x x V x, x V x = x + = x +(x+x )=(x +x)+x = +x = x x = x x = x =x =(+)x =x +x = x +x x = x ( )x = x =x =(+( ))x =x +( )x = x +( )x ( )x = x x x R V (I) () (4) (II) () (4) V K vector space V vector K scalor K C K R (I) x, y V x + y V () (x + y)+z = x +(y + z) (2) x + y = y + x (3) V x V x + = x (4) x V x + x = x V x x (II) x V, α K αx V () (α + β)x