선형대수

fundamentals: ; ; 1, 1, basis; ; complement, sum, direct sum; ; isomorphism ; quotient space ; quotient space ; duality: linear function, coordinate function linear function characterization; dual space, ; natural isomorphism between X and X ; annihilator (1); annihilator quotient space ; linear mapping: ; range null space, ; ; linear mapping algebra; adjoint mapping (1); annihilator+adjoint+range+null space; matrices: determinant: spectra: iteration problem eigenvalue; eigenvalue, eigenvector ; characteristic polynomial, Cayley-Hamilton; iteration problem generalized eigenvector ; spectral theorem Jordan form; minimal polynomial; innerproduct spaces:,, ;, Cauchy- Schwarz,, polarization identity; ; orthonormal basis, Gram-Schmidt; identification of X and X ; annihilator (2) orthogonal complement; adjoint mapping (2); characterization of euclidean motion; complex inner product; orthogonal matrix, orthogonal transform; linear groups; bilinear form, quadratic form; multilinear algebra; spectral theory: self-adjoint mappings and quadratic form; existence of eigenvector basis for self-adj. mappings; orthogonal diagonalization; minimax characterization of eigenvalues; calculus:, tarce, determinant ; exponential map; linear group lie algebra; examples; inequalities: positive definiteness; examples of positive definite quadratic form; Laplacian; i

ii...... motivation...,.. 1 2. 1 2.,... self-contained.....

iii. 1........,.. (algorithm)....... 1. 1 2. 1(linear transformation) 3. 4., 5. 2(quadratic form) typical., 1 (,,, ). 1 ax = b 1,2 y = ax + b, y = ax 2 + bx + c (, ). 1 2..

iv? 1 2.,, 1 R n (vector space) X.?. 1. R 2. 2., 1.. 1. upgrade, upgrade epsilon-delta, upgrade.?... shortcut.., (prerequisite).. 10.,... 1 2., 1, 1,, 2, 2.(.)

Contents i ii Chapter 1. 1 1.1. 1 1.2., 5 1.3. 10 15 Chapter 2. 17 2.1. 17 2.2. 22 2.3. Annihilator 25 28 Chapter 3. 29 3.1. 29 3.2. 33 3.3. 37 3.4. (Adjoint) 41 Chapter 4. 47 Chapter 5. 49 Chapter 6. : 51 Chapter 7. : 2 53 Chapter 8. Jordan : 55 Chapter 9. 57 vii

CHAPTER 1 1.1. 1.1.1. (scalar field)... (number).,..,,,, (scalar). (;field). 1 1.1. K (field) (+) ( ), K, K. : (1) k + h = h + k (2) k + (h + l) = (k + h) + l (3) 0 : k k+0 = k. (4) k h : k+h = 0 ( h k.) ( k h = kh ): (1) kh = hk (2) k(hl) = (kh)l 1. (;ring). (module). 1

2 1. (3) 1 : 1 0, k k1 = k. (4) 0 k h : kh = 1 ( h k 1 1.) k : (1) k(h + l) = kh + kl 1.1.? (1) R,. (2) C,. (3) Q,. (4) Q( 2) = {a + b 2 a, b },. (5) Z,. (6) Z( 2) = {m + n 2 m, n },. (7) {0, 1} : 0 + 0 = 1 + 1 = 0, 0 + 1 = 1 + 0 = 1, 0 0 = 0 1 = 1 0 = 0, 1 1 = 1. Z 2. 1.1.2.. n R n. n-tuple (x 1,..., x n ) (α 1,..., α n ) + (β 1,..., β n ) = (α 1 + β 1,..., α n + β n ), k(α 1,..., α n ) = (kα 1,..., kα n ).(k ) R n.. 1.2. K V (, ) + : V V V, : K V V, x, y, z V, α, β K, (V, +, ) V K (vector space), V (vector). (1) V +. +(x, y) x + y, x, y (sum). (2) commutativity: x + y = y + x ( )

1.1. 3 (3) associativity: x + (y + z) = (x + y) + z ( ) (4) zero: x, x + 0 = x 0 V. 0 (zero vector). (5) inverse: x, x + y = 0 y V. y x x (inverse). (6) V. (α, x) αx, x α(multiplication by α). (7) associativity: α(βx) = (αβ)x ( ) (8) 1 : 1x = x (9) distributivity(1): α(x + y) = αx + αy ( 1) (10) distributivity(2): (α + β)x = αx + βx ( 2) 1.2. (1) 0, 0 0 + 0. (2) x y, y y + x + y. K R (real vector space), K C (complex vector space).. 1.1.3.. R n 1.1. [0, 1] f : [0, 1] R X = C[0, 1]. α R f, g X f + g α f ( f + g)(t) := f (t) + g(t), (α f )(t) := α f (t) X. X. 2,,.. t [0, 1] 0(t) := 0 0 f X f + 0 = f. t [0, 1] ( f + 0)(t) = f (t) + 0(t) = f (t) + 0 = f (t) 2.

4 1.. t [0, 1] g(t) := f (t) g f + g = 0. ( f )(t) = f (t). 1.3.. 1.4.,. (1) [0, 1] 1 (C 1 ) C 1 [0, 1]. (2) t P. (3) t a 0 + a 1 t + a 2 t 2 + + a n t n + W. 1.5.? (1) R + : : x + y = xy, : kx = x k (2) (x, y) R 2 : : (x, y) + (x, y ) = (xx, yy ), : k(x, y) = (kx, ky) (3) (x, y) R 2 : : (x, y) + (x, y ) = (x + x + 1, y + y + 1), : k(x, y) = (kx + k 1, ky + k 1) (4) (1, y) : (1, y) + (1, y ) = (1, y + y ), : k(1, y) = (1, ky) 1.1.4... V X V, X V. (V, +, ) X V () 10. V X.. 1.1. (V, +, ) X V X. 1.6. V X V, X.

1.2., 5 1.7. (1) [0, 1] C [0, 1] C 1 [0, 1]. (2) t n P n P P. 1.8. (1) V U, W U W V. (2) V U, W U + W = {x + y x U, y W} V. (, U + W U, W (sum).) 1.2., x, y 0 y = αx.?,?,,,.??.... 1.2.1.,. 1.3. V x 1,..., x n 1 (α 1,..., α n ) (0,..., 0) α 1 x 1 + + α n x n = 0

6 1.. x 1,..., x n 1 (α 1,..., α n ) (0,..., 0). 3 1.9. x 1,..., x n V 1 : α 1 x 1 + + α n x n = 0, α 1 = = α n = 0. 1.10. R 2 (ξ 1, ξ 2 ) (η 1, η 2 ) ξ 1 η 2 = ξ 2 η 1. 1.11. x, y, z, x + y, y + z, z + x. 4 1.12. {x 1,..., x k } 1., {x 1,..., x k } 1 1. 1.2.2.. V x 1,..., x n α 1,..., α n α 1 x 1 + + α n x n x 1,..., x n 1(linear combination). 1.4. V x 1,..., x n 1 V x 1,..., x n V (span). 1.13. V x 1,..., x k, 1 S V. ( x 1,..., x k (subspace spanned by).) (1 ). 1.,,. 1.2. R 3 (1, 0, 0), (0, 1, 0), (0, 0, 1). R 3. 1 R 3. R 3 1. 1 R 3. (.) (1 ). 3, 1 1. 4, Z2.?

1.2., 7 1. 1.2. V x 1,..., x n V, V y 1,..., y k 1. k n. x 1,..., x n V V x 1,..., x n. y 1 = α 1 x 1 + + α n x n. y 1 0.(.) α 0. α i 0. x i x y 1 1. x i x y 1 V. k n, n 1 y 1,..., y n V. 5 k > n y 1,..., y k 1. k > n. k n. 1.5. V 1 (basis). 6 1.3. V x 1,..., x n.. x 1,..., x n 1 not-trivial 1 0. x i x 1. x i x V. x 1 x. 1.6. V, V (finite dimensional). 1.4. V V. 5. xi y 1 x x 1 y 1. y 1, x 2,..., x n. y 2 ( 0).,. β 2,..., β n y 2 = β 1 y 1 + β 2 x 2 + + β n x n 0. 0 y 2 = β 1 y 1 y 1. β j 0 j β 2, x 2 y 2. n. 6 (linear coordinate system).

8 1.. x 1,..., x n y 1,..., y k. 1.2 k n, n k. n = k.. 1.7. V (dimension) (minimal).. 1.2.3..? 1.5. V y 1,..., y j, V.. y 1,..., y j V y 1,..., y j j = n = dim V. y 1,..., y j V. (, j < n.) V y 1,..., y j. y j+1. y 1,..., y j+1. ( : α 1 y 1 + + α j+1 y j+1 = 0, α j+1 = 0., α j+1 0 y j+1 y 1,..., y j. α 1 y 1 + + α j y j = 0 α 1 = = α j = 0.) y 1,..., y j+1 V. y 1,..., y j+1,. V n y 1,..., y n. 1.6. (1) V U. (2). (3) V U V W V U W,., x V y U z W x = y + z. (, dim V = dim U + dim W.). (1) y 1 ( 0) U. 1.5 U.

1.2., 9 (2) (1) U = V. (3) U y 1,..., y j 1.5 V y 1,..., y j, y j+1,..., y n. y j+1,..., y n V W. x V x = α 1 y 1 + + α n y n (1.1). y = α 1 y 1 + + α j y j, z = α j+1 y j+1 + + α n y n. y U, z W, x = y + z.. x = y + z, y U, z W, x = y + z = (β 1 y 1 + + β j y j ) + (β j+1 y j+1 + + β n y n ). (1.1) α 1 y 1 + + α n y n = β 1 y 1 + + β n y n, y 1,..., y n α 1 = β 1,..., α n = β n y = y, z = z. n = j + (n j) dim V = dim U + dim W. 1.8. 1.6 W U (complement)., V U, W (direct sum). V = U W V W 1,..., W m x V x = y 1 + + y m, y j Y j V W 1,..., W m V = W 1 W m. 1.14. V V = W 1 W m dim V = dim W 1 + + dim W m. 1.15. V. U, W V. dim(u + W) = dim U + dim W dim(u W)

10 1. 1.3..,.. 1.3.1..,..,. (a, b). R 2.. (a, b) α = a + ib,. C. R 2 C.., R 2 (a, b) C a + ib, C c + id R 2 (c, d). ( ).... (a, b) + (c, d) = (a + c, b + d), k(a, b) = (ka, kb) (a + ib) + (c + id) = (a + c) + i(b + d), k(a + ib) = (ka) + i(kb) R 2 C... R 2, C.. U, V..?

1.3. 11 U, V.,., U, V U x V x. ( V U.) U x, y x, y,., x, y x + y, x, y x + y., U x + y V x + y., U kx V kx. U, V.. 1.9. f : X Y (one-to-one, injective) x, y X f (x) = f (y) x = y. f : X Y (onto, surjective) y Y x X y = f (x). f : X Y 11(one-to-one correspondence) f. 1.16. (1) 11 f. (2) f f 11. 1.10. (U, +, ) (V, +, ) 11 ϕ : U V ϕ (isomorphism) : x, y X, α, β K ϕ(αx + βy) = αϕ(x) + βϕ(y).,, U V (isomorphic). 1.17. : x, y X, α K ϕ(x + y) = ϕ(x) + ϕ(y), ϕ(αx) = αϕ(x). 1.18. (1) V (identity map) id V : V V, (id V (x) := x). (2). 1.7. K n V K n.. {x 1,..., x n } V. V x α 1 x 1 + + α n x n., x (α 1,..., α n ). V K n 11 x (α 1,..., α n )

12 1.., y = β 1 x 1 + + β n x n kx + hy = (kα 1 + hβ 1 )x 1 + + (kα n + hβ n )x n. 1.19.. (1) U = P 2, V = R 3 (2) U = R + (: x + y = xy, : kx = x k ), V = R 1 1.20.. 1.3.2. (Quotient Space). V U U W V = U W. U W. V U U. V U... R 2 U = {(α, 0) α }. U x. U y x U. 7 1.21. V = R 2 {(1, 0), (0, 1)}, {(1, 0), (1, 1)} (1) x V. (2) x. x. x. x.. V U. V x U (U x ). 7 y. y x., y x.

1.3. 13 U u x. x + U., x + U := {x + u u U}. U (coset). x U x + U {x}. 1.22. x + U y + U x y U. U.., (x + U) + (y + U) := (x + y) + U, α(x + U) := (αx) + U. 1.23. x, y. V U. (well-defined)., C 1, C 2, C 1 = x + U = x + U, C 2 = y + U = y + U C 1 + C 2. x x U y y U. (x + y) (x + y ) = (x x ) + (y y ) U. (x + y) + U = (x + y ) + U C 1 + C 2 (x + y) + U (x + y ) + U C 1 + C 2.. 1.8. V U... 8. 0 + U = U. 1.11. V U V/U U V (quotient space).

14 1. 1.3.3.. V/U U 0.. 1.3. V = R 5 U = {(0, 0, α 3 ) α 3 }. U C = (a 1, a 2, a 3 ) + U = (b 1, b 2, b 3 ) + U (a 1, a 2, a 3 ), (b 1, b 2, b 3 ) U a 1 = b 1, a 2 = b 2 a 3 b 3. U (a 1, a 2 )., U., U U.. 1.9. V U V. dim U + dim(v/u) = dim V. x 1,..., x j U. x j+1,..., x n V x 1,..., x n. x j+1 + U,..., x n + U V/U. x j+1 + U,..., x n + U 1. λ j+1 (x j+1 + U) + + λ n (x n + U) = (λ j+1 x j+1 + + λ n x n ) + U = 0 + U. (λ j+1 x j+1 + + λ n x n ) 0 U. λ 1,..., λ j λ j+1 x j+1 + + λ n x n = λ 1 x 1 + + λ j x j. λ 1 x 1 + + λ j x j λ j+1 x j+1 λ n x n = 0 x 1,..., x n V λ 1 = = λ n = 0., x j+1 + U,..., x n + U 1. x j+1 + U,..., x n + U V/U. V/U x + U. x 1 x = α 1 x 1 + + α n x n. x = α 1 x 1 + + α j+1 x j+1, x x U., x + U = x + U = α 1 (x 1 + U) + + α j+1 (x j+1 + U), x j+1 + U,..., x n + U V/U..

15 1.10. V U V U = V. 1.11. V U V V U V/U. (1). (2) 1, 1. (3). (4),. (5). (6).

CHAPTER 2....,,,.... 1 2. 1. 2.1. 1. 1 1. R 2 x, y ax + by..,.. 2.1.1... 2.1. V K. l : V K (linear function) x, y V α, β K l (αx + βy) = α l(x) + β l(y). 1 1 1 (linear functional). 20, () () 17

18 2. (linearity).. l (α 1 x 1 + + α n x n ) = α 1 l(x 1 ) + + α n l(x n )..,... V l, m : 2 (l + m)(x) := l(x) + m(x), (αl)(x) := α l(x). V K. V (dual space) V. 2.1. V K. 2.1. (1) V = R 2. f, g : V R f (x, y) := 3x + 2y, g(x, y) := 3x + 2y + 1. f. g. (2) C[0, 1] = { f : [0, 1] R f }.. l( f ) = 1 0 f (x) dx (3) C [0, 1] = { f : [0, 1] R, f C }. 0 < a < 1 d j f l( f ) := α j dx j (a) l. 2.2....... 2.

2.1. 19 2.1. V n, v 1,..., v n. x V.. x = α 1 x 1 + + α n x n (2.1) (1) x V α i = ϕ i (x) x ϕ i x. (2) a 1,..., a n x. l (x) = a 1 ϕ 1 (x) + + a n ϕ n (x) (2.2) (3) V 0 y l (y) 0 l. (4) V (2.2),. 3. (1). x, y V (2.1) x = α 1 x 1 + + α n x n, y = β 1 x 1 + + β n x n. ϕ i (x) = α i, ϕ i (y) = β i. αx + βy = (αα 1 + ββ 1 )x 1 + + (αα n + ββ n )x n ϕ i (αx + βy) = αα i + ββ i. ϕ i (αx + βy) = αϕ i (x) + βϕ i (y) ϕ i. (2).. (3). y 0 {x 1 := y} 1. n 1 x 2,..., x n V. ϕ i ϕ 1 (y) = ϕ(x 1 ) = 1 0, ϕ 1.. (4). l V, (2.1). l (x) = α 1 l(x 1 ) + + α n l(x n ) = l(x 1 )ϕ 1 (x) + + l(x n )ϕ n (x) (2.2). 3 (4)..

20 2. l = l(x 1 )ϕ 1 + + l(x n )ϕ n. l ϕ 1,..., ϕ n.,. 2.3. (1) V 0 y α, y(x) = α x V? (2) y z, z(x) = 0 x y(x) = 0. α y = αz. (: z(x 0 ) 0 α = y(x 0 )/z(x 0 ).) 2.1.2.. V. V ϕ 1,..., ϕ n. V.. 1.. α 1 ϕ 1 + + α n ϕ n = 0., x V 0. x i ( 0 ) 0 = α 1 ϕ 1 (x i ) + + α n ϕ n (x i ) = α i ϕ i (x i ) = α i α i 0.,. 2.2. V x 1,..., x n ϕ 1,..., ϕ n V. x 1,..., x n (dual basis). 2.3. V, V V. dim V = dim V. 2 x, y 1 ax + by. 1 a, b (a, b) 1., 1 R 2..

2.1. 21 2.1.3.. V V. (.)?.?.,.. V V V V. V V V V. V V. V V (double dual space). V V (V V ) V V??. V V.. 4 V. V x V ξ : ξ(l) := l(x) (2.3) ξ V. l, m V ξ., ξ(αl + βm) = (αl + βm)(x) = αl(x) + βm(x) = αξ(l) + βξ(m) x V ξ V Φ. Φ : V V, Φ(x)(l) = ξ(l) = l(x) Φ 11. x, y ξ. l V. l ξ(l) = l(x) = l(y) 0 = l(x) l(y) = l(x y) 4 V V. V V,.

22 2.. x y 0 l(x y) 0 l. l(x y) l 0 x y = 0., x = y. Φ : V Φ(V) = Ξ. 11..? x, y V Φ(αx + βy). l V., Φ(αx + βy)(l) = l(αx + βy) = αl(x) + βl(y) = αφ(x)(l) + βφ(x)(l).., Φ. Φ(αx + βy) = αφ(x) + βφ(y) Φ V Ξ. V Ξ. dim V = dim V = dim V. dim Ξ = dim V. 1.10 Ξ = V.,. 2.4. V, Φ V V.. Φ V V (natural, canonical)..,,. 5 2.2.... 5 Homology Category.

2.2. 23 2.2.1.. 3. 2.. n. 3 n. V = R 3. (1, 0, 0), (0, 1, 0), (0, 0, 1) x, y, z. V l, a, b, c v = (x, y, z) l(x, y, z) = ax + by + cz. x, y, z V, l (a, b, c). Φ(v). Φ(v) l = (a, b, c) Φ(v)(l) = l(v) = ax + by + cz = xa + yb + zc, (a, b, c) V Φ(v) (x, y, z).. (x, y, z) 1 l l (a, b, c) l(v) = ax + by + cz, l (a, b, c) () l(v) = Φ(v)(l) = xa + yb + zc = v(l)., (x, y, z) 1 (a, b, c).,,, v = (x, y, z) l = (a, b, c). v l Φ(v). V V., V = V. V V V V. V = V V V V = V V. (duality). 6 6... 19 20.

24 2. 2.2.2.. V. V V.... V x V l (l, x) = l(x) = Φ(x)(l). (l, x) l x., (, ) : V V R.. l x.. (αl + βm, x) = α(l, x) + β(m, x) (l, αx + βy) = α(l, x) + β(l, y) (2.4). ( ) (2.4) (bilinear function) (bilinear form). 2.2... V = R 2. v = (a, b) w = (c, d) (v, w) = ac + bd (, ) v, w (1).. ω 1 (v, w) = ac + 2bd, ω 2 (v, w) = 2ac bd, det(v, w) = ad bc. V V (, ). V l V x (l, x). V = V x = ξ V l (l, x) = (ξ, l).. V V V V.. V V Φ.

2.3. ANNIHILATOR 25 2.3. Annihilator 2.3.1. Annihilator. 0.... y = ax + b, y = x 2 1 ax y + b 2 x 2 y 0., 0.. 2.2. U V, U 0 l U annihilator U 0., U 0 = { l V l(x) = 0 for all x U}. 2.4. (1) U 0 V. (2) {0} 0 = V V 0 = {0} V. 2.5. V U U (V/U).. dim U + dim U 0 = dim V dim U 0 U (codimension) codim U.. U 0 (V/U) : l U 0 L (V/U). L{x} = L(x + U) := l(x) L (welldefinedness)., x, y L{x}. {x} = x + U = y + U = {y}. L{x} = l(x) L{y} = l(y)., x y U l U 0 l(x y) = 0. L{x} = l(x) = l(y) = L{y} L. l L (one-to-one). l, m U 0 l L m L. x V l(x) = L{x} = m(x) l m. l L.

26 2. l L (onto). L (V/U). L l U 0 : l(x) := L{x} l V. l U 0 x U, {x} = x + U = U = {0} l(x) = L{x} = L{0} = 0. l L. l L 11.. l, m U 0 l L, m M, (αl + βm){x} = αl{x} + βm{x} = αl(x) + βm(x) = (αl + βm)(x) αl + βm αl + βm.,. dim U 0 = dim(v/u) = dim V/U = dim V dim U. U annihilator. annihilator. U V U 0 V. U 0 annihilator U 00 V. V = V U 00 V.,. 2.6. V V Φ. U 00 = U. U U 00. x U l U 0 l(x) = 0., x U 0 0 V = V. x U 00.. dim U + dim U 0 = dim V = dim V = dim U 0 + dim U 00 dim U = dim U 00. U = U 00.

2.3. ANNIHILATOR 27 2.3.2....?.. 2.7. t 0,..., t n, n p m 0,..., m n : 7 1 p(t) dt = m 0 p(t 0 ) + + m n p(t n ) 0. n V = P n. a 0 + a 1 t + + a n t n (a 0,..., a n ) R n+1 dim P n = n + 1. l j l j (p) := p(t j ). ( p(t j ) p.) l j V. {l j } 1. λ 0 l 0 + + λ n l n = 0 V p. n 1 0 = λ 0 l 0 (p) + + λ n l n (p) = λ 0 p(t 0 ) + + λ n p(t n ) q k (t) = (t t 0 ) (t tk ) (t t n ) 8, t = t k t j 0, p = q k, 0 = 0 + + 0 + λ k q k (t k ) + 0 + + 0. q k (t k ) 0 λ k = 0. {l j } 1., dim V = n + 1 {l j } V. V l j 1. l(p) = 1 0 p(t) dt l p V. m 0,..., m n l = m 0 l 0 + m n l n 7 (quadrature formula). 8 (t tk ).

28 2... 2.5. : t 0,..., t n, n p m 0,..., m n. p (0) = m 0 p(t 0 ) + + m n p(t n ) 2.6. n = 2,. (1). (2). (3). (4).. (5). (6) Annihilator. (7) ().

CHAPTER 3 3.1. R n. (1).. 1?.. v = (x 1,..., x n ) T, A = (a ij ) m n A v., a 11 a 1n x 1 a 11 x 1 + + a 1n x n Av =...... =. a m1 a mn x n a m1 x 1 + + a mn x n R m. v 1 m...? (). V, W f : V W (mapping). 3.1.1.. 3.1. K V, W T : V W (linear mapping) T x, y V α K. (1) T(x + y) = T(x) + T(y). (Additivity) (2) T(αx) = α T(x). (Homogeniety) V = W T (linear transformation). 1 1. T(x + y) = Tx + Ty : V x y x + y T, x y T. 29

30 3. x T T(x) Tx. 3.1. (1) m n A T : R n R m, Tx := Ax. (2). (3) d/dt : P n P n. (4) V V K. (5) X V, (inclusion map) i : X V, i(x) := x. (6) X V, (quotient map) q. q : V V/X, q(x) := {x} = x + V 3.1.. 3.2. T : V W. (1) V U, T U T(U) = {Tx x U} W. (2) W U, T U T 1 (U) = {x V Tx U} V... 3.2. T : V W T V T(V) T (range space) R T. T.,...( (homomorphism).). V T T W. T(x + V y) = Tx + W Ty. x, y T.( T (+ V ) = (+ W ) T.).,., ( f + g) = f + g lim(a n + b n ) = lim a n + lim b n, a(b + c) = ab + ac., (?)..

3.1. 31, 0 T 1 (0) T ()(null space) N T. 3.3. T : V W. (1) T (onto) R T = W.. (2) T (one-to-one) N T = {0}. 3.1.2..... 2 R 2 l(x, y) = x + y. l 2 1., x + y = k l k., l x + y = 0. R 2 1 l {x + y = 0} l,. 2 1, 1.?,?.. 3.1. V T : V W. dim N T + dim R T = dim V. 3.2. V T : V W V/N T R T.. T T. T : V/N T R T W, T{x} = Tx. T. x, y V, k., T({x} + {y}) = T{x + y} = T(x + y) = T(x) + T(y) = T{x} + T{y} T(k{x}) = T{kx} = T(kx) = k T(x) = k T{x}.

32 3. T one-to-one. T{x} = T{y}, T Tx = Ty 0 = Tx Ty = T(x y) x y N T. {x} = {y} one-to-one. 2 Tx T{x} T T R T., T R T (onto map). T V/N T R T. 3.1.3... 3.3. T : V W. () dim W < dim V x( 0) V Tx = 0. () dim V = dim W Tx = 0 x = 0 R T = W. T. () dim V = dim W R T = W N T = {0}. T.. () dim R T dim W < dim V, dim N T = dim V dim R T > 0., N T 0. () N T = {0}. dim N T = 0. dim R T = dim V dim N T = dim V = dim W. 1.10 R T = W. () ()., 1. 3.4. V = R n, W = R m, A = (a ij ) m n. T(x) := Ax T : R n R m. () m < n ()1 a ij x j = 0 j x = (x 1,..., x n ) T 0. 2. 0 = T{x} = Tx, x N T. {x} = 0 T one-to-one.

3.2. 33 () m = n x = 0 ()1 a ij x j = y i j () y = (y 1,..., y n ) T x = (x 1,..., x n ) T. 3.4.... 3.2. V n(> 0). n S 1 = [a 1, b 1 ],..., S n = [a n, b n ], T : V R n. Tp = (p 1,..., p n ), p i = 1 b i a i bi a i p(t) dt N T = {0}. ( T.) : [a i, b i ] 0 p p i 0, p. p. p n. p n p 0. () T. n n. 3.5.. (1) T. (2) Tp n p. 3.2. R n.,?..

34 3. 3.2.1....... K V, W S, T : V W α K (S + T)(x) = S(x) + T(x), (αt)(x) = α T(x)., S + T αt. 3.5. S, T, α, S + T αt.. 3.6. V W L(V, W). T ( T)(x) := (Tx), 0(x) := 0. 3.6. 3.6. 3.2.2..... T : U V S : V W S T : 3 S T(x) = S(T(x)).. 3.7. : (1) R (S T) = (R S) T (:associativity) (2) (S 1 + S 2 ) T = S 1 T + S 2 T (:distributivity) (3) S (T 1 + T 2 ) = S T 1 + S T 2 (:distributivity). S T ST. 3..

3.2. 35. id V : V V id V (x) := x. T : V W S : U V T id V = T, id V S = S. 0(x) = 0. 0 S = T 0 = 0 T : V W, S : W V S T = id V, T S = id W, S T (inverse mapping).. : S 1, S 2 S 1 = S 1 id W = S 1 (T S 2 ) = (S 1 T) S 2 = id V S 2 = S 2. T T 1. 3.8. S : U V, T : V W. (1) N S N T S (2) R T S R T 3.7... 3.9. T : V W. 4 (1) T. (2) T.. T T 1 1 onto. T S : W V. S. y 1, y 2 W Sy i = x i, Tx i = y i, T(α i x i ) = α i y i, S(α i y i ) = α i x i. S(α 1 y 1 + α 2 y 2 ) = α 1 x 1 + α 2 x 2 = α 1 (Sy 1 ) + α 2 (Sy 2 ) S. 4. T : V W S1, S 2 : W V S 1 T = id V, TS 2 = id W T S 1 = S 2 = T 1..

36 3. T T 1 1 onto. {0} N T N S T = N idv = {0}. N T = {0} T 1 1. W R T R T S = R idw = W R T = W T onto. T. (;invertible). 3.8. (S 1 ) 1 = S., S, T ST, ST (ST) 1 = T 1 S 1. 3.2.3.. T T : V V V (linear transformation) (linear operator). V L(V). 5 L(V) A A n A 0 = id V, A n+1 = A A n. A n p(a) = α 0 id V +α 1 A + α 2 A 2 + + α m A m. V = C (R) f (t), D = d/dt D : V V. T = id +D 2 V. f (t) (T f )(t) = f (t) + f (t). f (t) = f (t) α sin t + β cos t. N T = {α sin t + β cos t} = Span{sin t, cos t}. 5 idv.

3.3. 37 3.3. 3.3.1.. R n 1. T(x) = Ax. (A m n.). ( ).. 1,. (R n ),. R n R m T. 6,,.. 3.10. T : R n R m m n A : x R n T(x) = Ax.. T(x) i t i (x), t i R n. 7 t i : t i = a i1 p 1 + + a in p n. p i R n., A = (a ij ) T(x) = (..., t i (x),... ) T = (..., a ij p j (x),... ) T = Ax.? T : V W T(x) = Ax. 8. V, W T : V W, V, W {x 1,..., x n }, {y 1,..., y m } V x x = ξ 1 x 1 + + ξ n x n, Tx Tx = η 1 y 1 + + η m y m.,. 6 loose.,. 7 qi : R m R i R m, q i. t i = q i T. 8 T(x) W Ax R m.

38 3. : 3.11. m n A = (a ij ) η i = a ij ξ j, T R n R m,... Tx j W a ij : m Tx j = a ij y i. Tx = T( ξ j x j ) = ξ j Tx j = ξ j a ij y i = ( a ij ξ j )y i i=1 j j j i i j. Tx = i η i y i. A x y T [T] y x.... Tx j y i 1 (η i ) (ξ j ) 1 transpose. 9..(.).... x, Tx. (.) x : x = ξ j x j = (x 1,..., x n ). = xξ. j ξ 1 ξ n 9 aij summation( ) i j.

3.3. 39, x = (x 1,..., x n ), ξ = [ξ 1,..., ξ n ] T. Tx = yη, A. Tx = (Tx j ) = (y i )(a ij ) = ya : Tx = T(xξ) = T(x)ξ = yaξ, Tx = yη η = Aξ. 3.9. t n P n d/dt : P n P n {1, t, t 2,..., t n }.. T : U V, S : V W, U, V, W x, y, z. T S S T. x U x = xξ Tx y [T] xξ y. S(Tx) z [S] z y[t] xξ y. (S T)x z [S T] z xξ ξ [S T] z xξ = [S] z y[t] xξ y. [S T] z x = [S] z y[t] y x.. 3.10. (1) V, id : V V. (2) T : V W T 1 : W V, V, W T T 1. 3.3.2... m n A R n T A (x) = Ax : R n R m, N A, R A., N A ={x x R n, Ax = 0} R A ={Ax x R n }

40 3. N A A, R A A (column space). A T., A T N A T A left null space, A T R A T A (row space). A (fundamental spaces). 3.3.3..,,.... T : V W V x, x W y, y Tx = y[t] y x, Tx = y [T] y x. x x A xa = x. B yb = y. TxA = yb[t] y x.,. y[t] y xa = yb[t] y x [T] y xa = B[T] y x 3.12. A B.. x x xa = x. x x x C = x. x = x C = xac. AC = I A. B. B 1 [T] y xa = [T] y x

. 3.4. (ADJOINT) 41 V = W T : V V.. 3.13. V x x xa = x T : V V : A 1 [T] x xa = [T] x x. A P, Q A 1 PA = Q P Q (similar). 3.4. (Adjoint) () ()..... 10 3.4.1.. T : V W, l W T (l) := l T : V T V l K. T l. V V., T W V. T (l). x V T. 11 T (l)(x) = (l T)(x) = l(t(x)) T : W V. : l 1, l 2 W α 1, α 2 K T (α 1 l 1 + α 2 l 2 )(x) = (α 1 l 1 + α 2 l 2 )(Tx) = α 1 l 1 (Tx) + α 2 l 2 (Tx) = (α 1 T (l 1 ) + α 2 T (l 2 ))(x) T. T T (;Adjoint). 10 (). 11. : (T (l), x) = (l, T(x)), (T l, x) = (l, Tx).

42 3. 3.11.. (1) 0 = 0 (2) (id) = id (3) (T + S) = T + S (4) (αt) = αt (5) (S T) = T S (6) (T 1 ) = (T ) 1 V V.. 3.14. T : V W. T = T. T : V W T : V W x V T x. T x W = W l W (T x)(l) = x(t l) = (T l)(x) = l(tx) = (Tx)(l). l W T x = Tx. 3.4.2..? V, W T V V, W W T T., V V. V x = {x 1,..., x n } V x φ = {φ 1,..., φ n }, W y = {y 1,..., y m } V y ψ = {ψ 1,..., ψ m }. T : V W T : W V. T A, T B., Tx = ya, T ψ = φb.. Tx j = a ip y p, T ψ i = b qi φ q p q. (T ψ i )x j. (T ψ i )x j = ψ i (Tx j ) = ψ i ( a pj y p ) = a pj ψ i (y p ) = a pj δ ip = a ij p p p., (T ψ i )x j = b qi φ q (x j ) = b qi δ qj = b ji q q a ij = b ji., A T = B. T T.

3.4. (ADJOINT) 43. R n l = (α 1,..., α n ) T, l(x) = l T x = α 1 ξ 1 + + α n ξ n = (α 1,..., α n ).. T(x) = Ax (T l)(x) (T l)(x) = l T (T(x)) = l T (Ax) = (l T A)x = (A T l) T x T A T. ξ 1 ξ n 3.4.3.. T T T.. m n A R A Ax. N A T A T l = 0 l, l T A = 0 l. l T (Ax) = l T Ax = (A T l) T x = 0x = 0. N A T l R A 0 (, R A )., R A N A T.. 3.15. T : V W. (R T ) 0 = N T, (R T ) 0 = N T. T = T,.. ( ) l (R T ) 0. y = Tx R T ( x V) l(y) = 0. x V, 0 = l(tx) = (T l)x. T l = 0., l N T. ( ) l N T., T l = 0. x V, 0 = (T l)x = l(tx)., y = Tx y l(y) = 0, l R T 0., l (R T ) 0. 3.16. dim R T = dim R T.

44 3.. dim N T + dim R T = dim W. annihilator dim(r T ) 0 + dim R T = dim W.,. 3.17. T : V W dim V = dim W : dim N T = dim N T.. 3.12. m < n l 1,..., l m n V 1. x V y j (x) = α j (j = 1,..., m) α 1,..., α m?, 1?? l i V R 1. (l 1,..., l m ) V R m. S. i = 1,..., m l i (x) = α i S(x) = (α 1,..., α m ). x v = (α 1,..., α m ) R m S range R S. v = (α 1,..., α m ) (N S ) 0.. v (N S ) 0 l N S l(v) = 0 [ S l = 0 l(v) = 0 ] [ [ z S l(z) = l(sz) = 0 ] l(v) = 0 ] [ S l = l S 1 0 l(v) = 0. ] (R m ) R m 1 l λ 1 k 1 + + λ m k m S l = l S = λ 1 l 1 + + λ m l m..

[ λ 1 l 1 + + λ m l m = 0 λ 1 α 1 + + λ m α m = 0 ] 3.4. (ADJOINT) 45. l i (x) = α i 1. m n 1 α i α i.., l i 1, l i 1 0 1 α i 1 0 α i x.

CHAPTER 4 47

CHAPTER 5 49

CHAPTER 6 : 51

CHAPTER 7 : 2 53

CHAPTER 8 Jordan : 55

CHAPTER 9 57