u V u V u u +( 1)u =(1+( 1))u =0 u = o u =( 1)u x = x 1 x 2. x n,y = y 1 y 2. y n K n = x 1 x 2. x n x + y x α αx x i K Kn α K x, y αx 1

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1 5 K K Q R C V V K K- 1) u, v V u + v V (a) u, v V u + v = v + u (b) u, v, w V (u + v)+w = u +(v + w) (c) u V u + o = u o V (d) u V u + u = o u V 2) α K u V u α αv V (a) α, β K u V (αβ)u = α(βv) (b) u V 1u = u (c) α, β K u V (α + β)u = αu + βu (d) α K U, v V α(u + v) =αu + αv V K-V 1) (c) o V o V 1) (c) o = o + o = o + o = o 1) (c) o V K- V 1) (d) u V u V u V 1) (d) u = u + o = u +(u + u )=(u + u)+u =(u + u )+u = o + u = u 46

2 u V u V u u +( 1)u =(1+( 1))u =0 u = o u =( 1)u x = x 1 x 2. x n,y = y 1 y 2. y n K n = x 1 x 2. x n x + y x α αx x i K Kn α K x, y αx 1 x 1 + y 1 x + y = x 2 + y 2, αx = αx 2..x n + y n αx n K K (m, n) M m,n (K) K- K n = M n,1 (K) [a, b] (a<b) C([a, b]) ϕ, ψ C([a, b]) ϕ + ψ C([a, b]) ϕ C([a, b]) λ R λϕ C([a, b]) (ϕ + ψ)(t) =ϕ(t)+ψ(t), (λϕ)(t) =λ ϕ(t) (aleqt b) C([a, b]) R K- V W V W V K-

3 ) W 2) w, w W w + w W 3) α K w W αw W K- V 1) V {o} V K- 2) W V K-V W [ ] 1) 2) W w W 0 K o =0 w W W K- V K-V W W K ) W V w W u =( 1)u V w K- V {v 1,v 2,,v r } { r } v 1,v 2,,v r K = α i v i α i K V K- [ ] W = v 1,,v r K V v, w v 1,,v r K v = r r α i v i, w = β i v i (α i,β i K) r v + w = (α i + β i )v i W α K r αv = (αα i )v i W

4 K- V K- W f : V W f V W K- 1) v, v V f(v + v = f(v)+f(v ) 2) α K v V 1) f(o) =ok- f V W 2) v V f( v) = f(v)k v V 0 v = o f(o) =f(0 v) =0 f(v) =o v =( 1)v f( v) =f(( 1)v) =( 1)f(v) = f(v) (m, n) A M m,n (K) f A : K n K m f A (x) =Ax f A K- K n K- K m K-f A A K K- V K- W K- f : V W Im(f) ={f(v) W v V }, Ker(f) ={v V f(v) =o} Im(f) W K-Ker(f) V K- Im(f), Ker(f) K- f [ ] Im(f) W w, w Im(f) w = f(v), w = f(v ) v, v V

5 50 5. f(v + v )==w + w w + w Im(f) α K f(αv) =αw αw Im(f) Im(f) W K- f(o) =o V Ker(f) Ker(f) V v, v Ker(f) f(v) = f(v ) = o f(v + v ) = o v + v Ker(f) α K f(αv) =αf(v) =o αv Ker(f) Ker(f) V K K- V K- W K- f : V W 1) f Im(f) =W 2) f Ker(f) ={o} [ ] 1) 2) f v Ker(f) f(v) =o f(o) =o f(v) =f(o) f v = o Ker(f) Ker(f) ={o} v, v V f(v) =f(v ) f(v v) =f(v ) f(v) =o v v Ker(f) Ker(f) v v = o v = v f K- V K- W K- f : V W f V W K- f : V W K- V,W K K- f : V W f f 1 : W V f 1 K- V,W K K- f : V W, g : W V f g =id W, g f =id V

6 K- V {v 1,v 2,v r } λ 1 v 1 + λ 2 v λ r v r = o λ i K λ 1 = λ 2 = = λ r =0 {v 1,v 2,,v r } K {v 1,v 2,,v r } K {v 1,v 2,,v r } K K- V {v 1,v 2,,v r } x 1 x 2 r x =.. Kr f(x) = x i v i V (5.1) x r K- f : K r V x 1 x 2 Ker(f) =.. Kr x 1v 1 + x 2 v x r v r = o x r ) {v 1,v 2,,v r } K (5.1) K- f K- V 1) {v 1,v 2,,v r } V K v i o (i =1, 2,,r) 2) v 1 V {v 1 } K v 1 o [ ] 1) v 1 = o λ 1 =1 λ 2 = = λ r =0 λ 1 v 1 + λ 2 v λ r v r = o {v 1,v 2,,v r } K 2) {v 1 } K v 1 o v 1 o 0 λ 1 K λ 1 v 1 = o λ 1 K v 1 = o {v 1 } K R 3

7 R- R 3 1) {v 1,v 2 } R 3 R {v 1,v 2 } 2) {v 1,v 2,v 3 } R 3 R {v 1,v 2,v 3 } K K n r v 1,v 2,,v r v j K n a 1j a 2j v j =. (a ij K). a nj K λ 1,λ 2,,λ r λ 1 v 1 + λ 2 v λ r v r = o λ j a 11 λ 1 + a 12 λ a 1r λ r =0 a 21 λ 1 + a 22 λ a 2r λ r =0.... a n1 λ 1 + a n2 λ a nr λ r =0 {v 1,v 2,,v r } K λ 1 = λ 2 = = λ r = rank(v 1,v 2,,v r )=r {v 1,v 2,,v r } K n K rank(v 1,v 2,,v r )=r (v 1,v 2,,v r ) v 1,v 2,,v r (n, r) K- V r K r V K dim K V r dim K V = dim K V < K- V K-

8 K- V dim K V = ) K- V dim K V K- K n K n dim K K n = n [ ] K n n e 1 = 0,e 2 = 0,,e n = (e 1,e 2,,e n )=I n rank(e 1,e 2 m,e n )= n {e 1.e 2..e n } K dim K K n n r >n K n r v 1,v 2,,v r rank(v 1,v 2,,v r ) n<r {v 1,v 2,,v r } K dim K K n = n K- V,W K- f : V W 1) f dim K V dim K W 2) f dim K V dim K W f K- dim K V = dim K W [ ] 1){w 1,w 2,,w r } W K f f(v i )=w i v i V {v 1,v 2,,v r } V K α 1 v 1 + α 2 v α r v r = o (α i K) K- f f(o) =o α 1 w 1 +α 2 w 2 + +α r w r = o α 1 = α 2 = cdots = α r =0 dim K V dim K W

9 ) {v 1,v 2,,v r } V K w i = f(v i ) {w 1,w 2,,w r } W K α 1 w 1 + α 2 w α r w r = o (α i K) f K- f(α 1 v 1 + α 2 v α r v r )=o = f(o) f α 1 v 1 + α 2 v α r v r = o α 1 = α 2 = cdots = α r =0 dim K V dim K W K- V {v 1,v 2,,v n } {v 1,v 2,,v n } V K 1) {v 1,v 2,,v n } K 2) V = v 1,v 2,,v n K K- V o V K [ ] dim K V = n n 1 K {v 1,v 2,,v n } V v V {v 1,v 2,,v n,v} K α 1 v 1 + α 2 v α n v n + α n+1 v = o α i K i α i 0 α n+1 =0 {v 1,v 2,,v n } K α n+1 0 v =( α 1 /α n+1 )v 1 + +( α n /α n+1 )v n v v 1,v 2,,v n K V = v 1,v 2,,v n K {v 1,v 2,,v n } V K K- V {v 1,v 2,,v n } x 1 x 2 n f : K n. x i v i V x n

10 f K- K n V K- Im(f) = v 1,v 2,,v n K, x 1 x 2 Ker(f) =. x 1v 1 + x 2 v x n v n = o x n ) f V = v 1,v 2,,v n K 2) f {v 1,v 2,,v n } K 3) f {v 1,v 2,,v n } V K {v 1,v 2,,v n } V K K= K n K- V K dim K V = n K- V V K K- V {v 1,v 2,,v n,v n+1 } (n 1) 1) v 1,v 2,,v n,v n+1 K = v 1,v 2,,v n K, 2) v n+1 = α 1 v 1 + α 2 v α n v n α i K [ ] 1) 2) v n+1 v 1,v 2,,v n+1 K = v 1,v 2,,v n K 2) 1) v 1,v 2,,v n K v 1,v 2,,v n,v n+1 K v v 1,,v n,v n+1 K n+1 v = λ i v i (λ i K) = n (λ ; + λ n+1 α i )v i v v 1,,v n K

11 K- V {v 1,v 2,,v n } 1) V K {v 1,v 2,,v r } V K V K {v 1,,v r,v r+1,,v n } 2) V = v 1,v 2,,v m K (v i V ) V K {v 1,v 2,,v m } V K [ ] 1){v 1,v 2,,v r } V K {v 1,,v r,,v n } V K {v 1,,v r,,v n } V K V = v 1,,v r,,v n K v V {v 1,,v r,,v n,v} K α 1 v α r v r + + α n v n + α n+1 v = o α i K i α i 0 α n+1 =0 {v 1,,v r,,v n } K α n+1 0 v =( α 1 /α n+1 )v 1 + +( α r /α n+1 )v r + +( α n /α n+1 )v n v v 1,,v r,,v n K 2) {v 1,v 2,,v m } K V S V K S = {v 1,v 2,,v n } (n m) K K α 1 v 1 + α 2 v α n v = o α i K i α i 0 α n 0 v n =( α 1 /α n )v 1 +( α 2 /α n )v n + +( α n 1 /α n )v n V = v 1,v 2,,v n 1 K S

12 n K- V n {v 1,v 2,,v n } 1) {v 1,v 2,,v n } V K 2) {v 1,v 2,,v n } K 3) V = v 1,v 2,,v n K [ ] 1) 2), 3) 2) 1) {v 1,v 2,,v n } K ) {v 1,v 2,,v n } V V K n = dim K V {v 1,v 2,,v n } V K 3) 1) ) {v 1,v 2,,v n } V K n = dim K V {v 1,v 2,,v n } V K K- K- V,W W V dim K V = dim K W< V = W [ ] dim K W = dim K V = n {w 1,w 2,,w n } W K {w 1,w 2,,w n } V K V = w 1,w 2,,w n K = V K- K n K- K n n {v 1,v 2,,v n } K n K n (v 1,v 2,,v n ) det(v 1,v 2,,v n ) 0 [ ] {v 1,v 2,,v n } K- K n K {v 1,v 2,,v n } K {v 1,v 2,,v n } K rank(v 1,v 2,,v n )=n (v 1,v 2,,v n ) n rank(v 1,v 2,,v n )=n (v 1,v 2,,v n )

13 K- V K- W K- f : V W V K Im(f) K dim K Im(f) = dim K V dim K Ker(f) [ ] Im(f) ={o} dim K Im(f) =0 Ker(f) =V Ker(f) ={o} f V Im(f) K dim K Im(f) = dim K V Im(f) {o} Ker(f) {o} V K {v 1,v 2,,v n } V = v 1,v 2,,v n K Im(f) = f(v 1 ),f(v 2 ),,f(v n ) K ) Im(f) K Im(f) K {w 1,,w r } r = dim K Im(f) w i Im(f) w i = f(v i ) v i V Ker(f) K {u 1,,u s } s = dim K Ker(f) {v 1,,v r,u 1, cdots, u s } V K α i,β j K α 1 v α r v r + β 1 u β s u s = o (5.2) f s β ju j Ker(f) f(v i )=w i r α iw i = o {w 1,,w r } α 1 = = α r =0 (5.2) s β ju j = o {u 1,,u s } β 1 = = β s =0 {v 1,,v r,u 1,,u s } K v V f(v) Im(f) f(v) = r α iw i α i K u = v r α iv i V u Ker(f) u = s β ju j β j K v = r α i v i + s β j u j v 1,,v r,u 1,,u s K {v 1,,v r,u 1,,u s } V K dim K V = r + s = dim K Im(f) + Ker(f)

14 (m, n) A M m,n (K) K- f A : K n K m f A (x) =Ax Ker(f A ) = Ker(A) (m, n) A M m,n (K) K- f A dim K Ker(f A )=n rank(a), dim K Im(f A ) = rank(a) [ ] rank(a) =r dim K Ker(f A )=n r K- f A : K n K m dim K Im(f A )=r Ker(f A ) = Ker(A) = q r+1,q r+2,,q n K Q =(q 1,q,,q n ) n {q 1,q 2,,q n } K n K {q r+1,q r+2,,q n } K {q r+1,q r+2,,q n } Ker(f A ) K dim K Ker(f A )=n r (m, n)- A M m,n (K) Ker(F A ) Im(f A ) K rank(a) =r [ ] I r 0 PAQ = 0 0 m P n Q Q =(q 1,q 2,,q n ), (q i K n ) {q r+1,,q n } Ker(f A ) K Im(f A ) [ ] f A (x) =Ax = P 1 I r 0 Q 1 x (x K n ) 0 0 y = Q 1 x Q 1 x K n y K n P 1 =(v 1,v 2,,v m ) (v j K m )

15 60 5. y y 1,,y n [ ] f A (x) =P 1 I r 0 y = y 1 v y r vr 0 0 Im(f A )= v 1,,v r K P 1 {v 1,,v r } K {v 1,,v r } Im(f A ) K (m, n)- A M m,n (K) [ ] I r 0 PAQ = (r = rank(a)) 0 0 P, Q P 1 =(v 1,,v m ), Q =(q 1,,q n ) (v j K m,q i K n ) {v 1,,v r } Im(f A ) K {q r+1,,q n } Ker(f A ) K P, Q K- K n {v 1,v 2,,v m } dim K v 1,v 2,,v m K = rank(v 1,v 2,,v m ) (v 1,v 2,,v m ) {v 1,v 2,,v n } (n, m) [ ] A =(v 1,v 2,,v n ) M n,m (K) Im(f A )= v 1,v 2,,v m K K- V,W K- f : V W V K [v 1,v 2,,v n } K- x 1 x 2 n ϕ : K n V (. x j v j ) x n

16 W K {w 1,w 2,,w m } K- y 1 y 2 m ψ : K m W (. y i w i ) K- f : V W K- V,W K n, K m j =1,,n f(v j ) W {w 1,,w m } K f(v j )= y m m a ij w j (a ij K) v = n x jv j V f(v) = n x j f(v j )= n x j m a ij w i (5.3) m n = a ij x j w i (5.4) a 11 a 12 a 1n a 21 a 22 a 2n A = M mn(k) (5.5) a m1 a m2 a mn K- f A (x) =Ax (x K n ) f ϕ(x) =ψ f A (x) (x K n ) K n ϕ V f A K m ψ f W. K- f : V W f A : K n K m (5.5) {v 1,,v n }, {w 1,,w m } f f V,W K- f : V W K- U K-

17 62 5. g : U V f g U W K- U K- {u 1,,u l } f,g f g k =1,,l g(u k )= n m b jk v j (f g)(u k )= c ik w i (b jk,c ik K) (5.6) g, f g B = (b jk ) j,k M nl (K), C = (c ik ) i,k M ml (K) (f g)(u k )=f(g(u k )) = = n b jk m n b jk f(v j ) m n a ij w i = a ij b jk w i (5.6) c ik = n a ijb jk K- g : U V, f : V W A, B f g : U W AB

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