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

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1 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 = αx + βx (2) α(x + y) =αx + αy (3) α(βx) =(αβ)x (4) x = x (2) (3) (4) x + = + x = x, x +( x) =( x)+x = V x, y x y = x +( y), V = + = =

2 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 n R n x R n x 2 = x,x 2,,x n R x n R n x y x + y x αx x 2 y 2 x 2 + y 2 x 2 αx 2 + =, α = x n y n x n + y n x n αx n R n R (I)(II) = t (,,,), t (x,x 2,,x n ) t ( x, x 2,, x n ) R n n Euclidean space 2 R 2 3 R 3 R m n M(m, n; R) a a n M(m, n; R) = a ij R, i m, j n a m a mn M(m, n; R) R 2

3 2 2 R R F f,g F α R f + g αf x R (f + g)(x) =f(x)+g(x), (αf)(x) =αf(x) F R (I)()(2) f (x) f x R f(x) = f(x) f (II) f = f, ( )f = f f f f (x) =(x) f f = f F f x f(x) f(x) =(x) 3 3= 3 3= f f = F R 2 P P = {{a n } n=,2, a n R,n N} {a n }, {b n } P α R {a n } + {b n } α{a n } {a n } + {b n } = {a n + b n }, α{a n } = {αa n } P R V W V W V vector subspace W ( φ) (I)(II) x, y W = x + y W, x W, α R = αx W 3

4 W V (I)()(2) (II)() (4) (I)(3)(4) V W x W x V x W 2 α = x = W α = ( )x = x W V {} V 3 () n M(n; R) =M(n, n; R) S, A, U M(n; R) (2) F 2 C = {f F f(x) R } C = {f C f(x) R } C F C C F C = {f F f(x) } F (3) P 2 G λ = {{a n } P a n+ = λa n (n N)} G λ P A λ = {{a n } P a n+ = a n + λ (n N)} λ P a,a 2,,a n (a,a 2,,a n ) x (,,,) R R n x 2 = x,x 2,,x n R x n 4

5 x Hc n = x 2 R n a x + a 2 x a n x n = c x n H n c Hn c x y x 2 y 2, H c n α R x n y n a (x + y )+a 2 (x 2 + y 2 )+ + a n (x n + y n )=2c a (αx )+a 2 (αx 2 )+ + a n (αx n )=α(a x + a 2 x a n x n )=αc Hc n c = H 3 c R3 H n c Rn hyperplane H 2 c R 2 H n Hc n (c ) 4 R R 3 R 3 x x () W = y x + y + z = (2) W 2 = y x, y, z z z x (3) W 3 = y xyz = z x (4) W 4 = y z x 2 = y 3 = z 4 x (5) W 5 = y x 2 + y 2 + z 2 = z x (7) W 7 = y x =y = z z x (6) W 6 = y x = y = z z x (8) W 8 = y x 2 + y 2 + z 2 < z 5

6 5 V R a, a 2,,a m V, λ,λ 2,,λ m R λ a + λ 2 a λ m a m a, a 2,,a m linear combination a, a 2,,a m V S S = {λ a + λ 2 a λ m a m λ,λ 2,,λ m R} S V 5 S a, a 2,,a m span[a, a 2,,a m ] G(a, a 2,,a m ), {{a, a 2,,a m }}, [a, a 2,,a m ] 6 R 3 a =, a 2 =, a 3 = λ + λ 3 µ span[a, a 2, a 3 ]= λ 2 λ 3 λ,λ 2,λ 3 R = µ 2 λ 2 + λ 3 µ 2 x = y y + z = = span[a, a 2 ] z µ,µ 2 R a, a 2, a 3 a, a 2 a 3 = a + a 2 a, a 2 a 3 span[a, a 2 ] R 3 a a 2 a, a 2 6 V R V a, a 2,,a m span[a, a 2,,a m ] b 6

7 span[a, a 2,,a m, b] = span[a, a 2,,a m ] V a, a 2,,a m m a, a 2,,a m linearly dependent linearly independent () a, a 2,,a m λ a + λ 2 a λ m a m = λ,λ 2,,λ m (λ,λ 2,,λ m ) (,,,) (2) a, a 2,,a m λ a + λ 2 a λ m a m = (λ,λ 2,,λ m )=(,,,) 7 R 2 R 2 a, b a = 2, b = 7+k 7 k 4 2α +(7+k)β =, αa + βb = 4 7+k (7 k)α +4β =, 48α + (28 + 4k)β =, (k 2 )α = (49 k 2 )α + (28 + k)β = k ± α = β = k = 3α +2β = k = 2α + β = (α, β) (, ) α, β αa + βb = k = 2a +3b =, k = a +2b = a, b k ± k = ± a, b R 2 αa + βb = α a = β b, β α b = α a (α, β) (, ) a, b β a, b a, b a, b R 2 a, b R 2 a, b R 2 a, b R 2 7

8 7 R 2 a = a, a 2 = b, c d () a, a 2, ad bc (2) ad bc, R 2 e, e 2 a, a 2 8 R 3 R 3 a, b, c k a =, b =, c = αa + βb + γc = γk =,α = β,α + β + γ = k γ =, α = β = k = γ = 2α, α = β, (α, β, γ) (,, ) α, β, γ αa + βb + γc = a + b 2c = a, b, c k k = a, b, c R 3 αa + βb + γc = α a = β α b γ c, α a b c a b c a, b, c β,γ a, b, c R 3 a, b, c R 3 a, b, c R 3 a, b, c R 3 8 R () a =, b = 3, c = 3 (2) a =, b =, c = (3) a =, b =, c = (4) a =, b =, c =, d = 8

9 2 (5) a =, b =, c = 3 k (k ) 9 R n R n a, a 2,,a n λ a + λ 2 a λ n a n = (λ,λ 2,,λ n ) = (,,,) a, a 2,,a n n A n λ = t (λ,λ 2,,λ n ) λ Aλ = λ = (A) 9 (A) n A n a, a 2,,a n det A = (B) R n k a,,a k n P a,,a k P a,,pa k V R V a, a 2 aa + ba 2,ca + da 2 a, b, c, d aa + ba 2,ca + da 2 α(aa + ba 2 )+ β(ca + da 2 )= α = β = a, a 2 αa + βc = α, β α = β = αb + βd = ad bc V R (A) a, b, c V () a, b, c 2 (2) a + b, b + c, c + a (3) a, a + b + c, a b c (4) x = a + b 2c, y = a b c, z = a + c (5) u = a + b 3c, v = a +3b c, w = b + c 9

10 (B) a, a 2,,a m V, c 2,c 3,,c m R a = a + c 2 a 2 + c 3 a c m a m, a, a 2,,a m a, a 2,,a m (C) a, a 2,,a k V a, a 2,,a k V c a + c 2 a c k a k = c a + c 2 a c k a k = c = c,c 2 = c 2,,c k = c k, 9(B) A rank A =(A ) =(A ) n A =(a, a 2,,a n ) rank A = n a, a 2,,a n n A =(a, a 2,,a n ) () A n (2) rank A = n (3) A n a, a 2,,a n (4) det A (5) x R n Ax = b b R n (6) x R n Ax = (7) A

11 V a, a 2,,a n V () span[a, a 2,,a n ]=V (2) a, a 2,,a n a, a 2,,a n V basis a, a 2,,a n R R n = x x 2 x n x,x 2,,x n R e =, e 2 =,,e n = R n x = x x 2 x n () x = x e + x 2 e x n e n (2) x e + x 2 e x n e n = x x 2 x n = e, e 2,,e n e, e 2,,e n R n () R n a =, a 2 =,,a n =

12 (2) R C (3) C C (4) R V a, b a + b, a b 2 (4) V n V n dim V = n (2)(3) R C 2 C C C V V x P f (x) =,f (x) =x, f 2 (x) =x 2,,f n (x) =x n, P 2 x () W = y R3 x = y = z z x x 2 (2) H = R 4 x x + x 2 + x 3 + x 4 = 3 x 4 (3) R C 2 (4) C C 2 n V W a, a 2,,a k n k V b, b 2,,b n k a, a 2,,a k, b, b 2,,b n k V V k V 2

13 (I) A, B M(m, n; R) A + B M(m, n; R) ()(2) (3) m n O m,n, (4)A A (II) A M(m, n; R) α R αa M(m, n; R) () (4) 2 (I)() ({a n } + {b n })+{c n } = {a n + b n } + {c n } = {(a n + b n )+c n } = {a n +(b n + c n )} = {a n } + {b n + c n } = {a n } +({b n } + {c n }) (2) (3) {z n } : z =,z 2 =, (4){a n } {a n } = { a n } {a n } {z n } = {}, {a n } = {a n } 3 () S, A, U M(n; R) S M(n; R) A, U (2) C C F C F C C f F C f(x) ( x R), f F C F C F (3) G λ P {a n }, {b n } G λ a n+ + b n+ = λa n + λb n = λ(a n + b n ) {a n } + {b n } = {a n + b n } G λ α R αa n+ = αλa n = λ(αa n ) α{a n } = {αa n } G λ G λ A λ P {a n }, {b n } A λ a n+ + b n+ = a n + λa n + b n + λ = a n + b n +2λ λ {a n } + {b n } = {a n + b n } A λ, λ = {a n } + {b n } A α R αa n+ = α(a n + λ) =αa n + αλ λ α{a n } = {αa n } A λ, λ = α{a n } A A P A λ (λ ) 2 4 (4)(6) W 4 = t 3 t R, W 6 = t t R 4 3

14 ()(3)(5)(7) (2)(8) x x y α R α y z z a = 2 (2) a W 2 3a W 2 (8) a W 8 2a W 8 5 S x, y x = λ a + + λ m a m, y = µ a + + µ m a m x + y =(λ + µ )a + +(λ m + µ m )a m S α R αx =(αλ )a + +(αλ m )a m S 6 span[a, a 2,,a m, b] span[a, a 2,,a m ] x span[a, a 2,,a m ] x = λ a + λ 2 a λ m a m = λ a + λ 2 a λ m a m +b x span[a, a 2,,a m, b] x span[a, a 2,,a m, b] b span[a, a 2,,a m ] b = λ a + λ 2 a λ m a m x = µ a + + µ m a m + µ m+ b =(µ + µ m+ λ )a + +(µ m + µ m+ λ m )a m x span[a, a 2,,a m ] span[a, a 2,,a m, b] span[a, a 2,,a m ] span[a, a 2,,a m, b] = span[a, a 2,,a m ] aα + bβ =, 7 () αa + βa 2 = α, β cα + dβ = a b α = c d β a, a 2 ad bc = α = β = ad bc 4

15 ad bc a c a, a 2 b α = d β 9(A) R n (2) αa +βa 2 = a b α = = e α = c d β β d e = ad bc c ad bc (da ca 2 ) e 2 = ad bc d c b = a ad bc ( ba + aa 2 ) α + β +5γ =, 8 () αa + βb + γc = α +3β +3γ =, 3 β +2γ =, β =2γ 2 α =9γ 6γ = γ = α = β = (2) (3) (4) (5)k k = 9 (A) α,α 2,,α n R, α a + + α n a n = A α α n = ( ) det A, A, (α,,α n ) ( ) α = = α n = a,,a n det A =, ( ) (α,,α n ) (,,) a,,a n (B) α (P a )+ + α k (P a k )=P(α a + + α k a k )= P α a + + α k a k = a,,a k α = = α k = P a,,pα k (A) α, β, γ R () a, b, c 2, a, b,, a = αb a, b, c (2) α(a + b)+β(b + c)+γ(c + a) =(α + γ)a +(α + β)b +(β + γ)c = a, b, c, α + γ = α + β = β + γ =, α = β = γ = (3) αa + β(a + b + c)+γ(a b c) =(α + β + γ)a +(β γ)b +(β γ)c = a, b, c, α = 2β,γ = β 5

16 (4) αx + βy + γz =(α + β + γ)a +(α β)b +( 2α β + γ)c =, a, b, c, α + β + γ = α β = 2α β + γ =, α = β = γ = x, y, z (5) αu + βv + γw =(α + β)a +(α +3β + γ)b +( 3α β + γ)c =, a, b, c, α + β = α +3β + γ = 3α β + γ =, β = α, γ =2α, u, v, w (B) b,b 2,,b m F, b a + b 2a b m a m = b a +(b c 2 + b 2 )a 2 + +(b c m + b m )a m = a, a 2,,a m, b = b c 2 + b 2 = = b c m + b m =, b = b 2 = = b m = a, a 2,,a m (C) c a + c 2 a c k a k = c a + c 2a c ka k (c c )a +(c 2 c 2 )a 2 + +(c k c k )a k = a, a 2,,a k c c = c 2 c 2 = = c k c k = c = c,c 2 = c 2,,c k = c k c = c 2 = = c k = a, a 2,,a k x () R n x 2 x = e, e 2,,e n x n n x = x i e i a i = e i, = (i =,,n) i= ( n n n nk= ) x k x = x i a i + x k = i= k= i= n x i a i a, a 2,,a n n n n x i a i = x i = x i e i i= i= i= n x i = x = x 2 = = x n x = x 2 = = x n = a, a 2,,a n i= a, a 2,,a n R n (2) C x α,α 2 R e =, e 2 = i x = α e + α 2 e 2 e, e 2 =,i R C (3) C x α = x C e = x = αe e = C C 6

17 (4) V x α, β R x = αa + βb λ = α + β,µ = α β αa + βb = λ(a + b) +µ(a b) x 2 2 a + b, a b λ(a + b)+µ(a b) = a, b λ + µ = λ µ = λ = µ = a + b, a b a + b, a b 2 () a = W = {ta t R} W a R3 (2) a =, b =, c = H = {ra + sb + tc r, s, t R} H a, b, c R 4 3 (3) e =, e 2 =, e 3 = i, e 4 = R C 2 i z R a, b, c, d z = ae + be 2 + ce 3 + de 4 C 2 z 2 z 2 e, e 2, e 3, e 4 4 (4) e =, e 2 = C C 2 z C a = z,b= z 2 z = ae + be 2 C 2 e, e 2 2 z 2 z 2 7

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