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C R Z Q a11x1 + + a1nxn = b1.....................................

a x1 + + amnxn = bm m1 aij, bj (1,..., n) i = 1,..., m ai11 + + ainn = bi.

a11 a1n...............

am1 amn a11 a1n b1....................

am1 amn bm a11 a1n b1 .

am1 amn bm i j (1,..., n) i (1,..., n) i (ai1 + aj1)1 + + (ain + ajn)n = (ai11 + + ainn) + (aj11 + + ajnn) = bi + + bj.

(1,..., n) i j - i (i, ki) ki+1 > ki aj,k = 0 j = i i A A = 0 A = 0 m ai1 i = 1 a-1 a11 = 1 11 m = 1 m > 1 m - 1 i > 1 i ai1a-1 ai1 = 0, i > 1 11 B A B B 1 < k2 < k3 < a1,k 2 a1,k 3 xi x1,..., xi-1 xi (0,..., 0, 1).

0x1 + + 0xn = 1, A x1,..., xr xr+1,..., xn +a1,r+1xr+1 + + a1nxn = b1 x1 x2 +a2,r+1xr+1 + + a2nxn = b2...........................................

xr +ar,r+1xr+1 + + arnxn = br x1 = b1 - a1,r+1xr+1 - - a1nxn x = b2 - a2,r+1xr+1 - - a2nxn 2......................................

x = br - ar,r+1xr+1 - - arnxn.

r b1 = = bm = 0 n m Mat(n m) n m A Mat(n m) A = Anm.

Anm = (aij), Bnm = (bij) A + B = (aij + bij) Anm = (aij) A, B, C Mat(n m), A + B = B + A A + (B + C) = (A + B) + C A + 0 = A A A -A A + (-A) = 0 (A + B) = A + B ( + )A = A + A ()A = (A) 1A = A A = (aij), B = (bij) A + B = (aij + bij) = (bij + aij) = B + A Anm = (aij), Cmk = (cst).

D = AC Mat(n k) = (dis) i = 1,..., n, s = 1,..., k dis = ai1d1s + + aindns (AC)F = A(CF ) A Mat(n m), C Mat(m k), F Mat(k l).

A = Anm = (aij), C = Cmk = (cst), F = Fkl = (ftq).

D (i, q) (AC)F = DF k di,f,q = =k n ai,d,f,q.

=1 =CF = U = (ui,) Mat(m l), (i, q) A(CF ) = AU n ai,u,q = =n k ai,d,f,q.

=1 = (AB) = (A)B = A(B).

A(B + C) = AB + AC, (A + U)V = AV + UV.

A = (aij), B = (bij), C = (cij) (i, j) A(B + C) aik(bkj + ckj) = aikbkj + aikckj, AB + AC A(B+C) AB+AC A = (aij) Mat(n) tr A a11 + +ann A, B Mat(n) tr(AB) = tr(BA) A = (aij), B = (bij). (i, i) AB n aijbji j=n tr(AB) = aijbji.

i,j=n n tr(BA) = bstats = atsbst = tr(AB).

s,t=1 s,t= ij i = j i = j E = En Mat(n) (i, j) ij A Mat(n m). EnA = A = AEm A = (aij) (i, j) EnA n ikakj = iiaij = aij, k= EnA = A.

t A Mat(n m) A = A Mat(m n) (i, j) aji A t t t (A + B) =tA +tB, (A) = tA, (AC) =tCtA.

t (AC) (i, j) ajkcki = ckiajk k k t CtA.

Eij Mat(n m) Eij (s, t) sitj (i, j) t Eij = Eji A = (aij) A = aijEij.

i,j EijErs = jrEis.

(u, v) EijErs 1, u = i = p = j, v = s;

(uipj)(pivs) = p A = (ars) Mat(n m) EijA = aj1Ei1 + + ajmEim, AEij = a1iE1j + + anjEni.

A Mat(n m) i j (En + Eij)A (En + Eij)A = EnA + EijA = A + (aj1Ei1 + + ajmEim) = arsErs + (aj1)Ei1 + + (ajm)Eim = rs arsErs + (ais + ajs)Eis.

r =i,s i,s A Mat(n m) i j A(Em + Eji) Di() = En + ( - 1)Eii Mat(n) A Mat(nm) i Di()A ADi() A = (aij) Di() (s, t) st + ( - 1)isit Di()A (p, q) (pr + ( - 1)ipir)arq = prarq + ( - 1) ipirarq = r r r apq, i = q;

apq + ( - 1)ipaiq = aiq, p = i.

AX = b A x X = xn b b = bm f : X Y.

x, y X f(x) = f(y) x = y y Y x X f(x) = y y Y f-1(y) = {x X|f(x) = y}.

|f-1(y)| 1 y Y |f-1(y)| 1 y Y |f-1(y)| = 1 y Y f : X Y, g : Y Z gf : X Z 1X : X X f-1 : Y X f 1Y f = f1X = f f f-1f = 1X ff-1 = 1Y ;

f f X X X Xn = {1, 2,..., n} n Xn Sn n Sn Xn = {i1,..., in} i1... in = (i1)... (i1) j1... jn =.

i1... in j1... jn = (i1)... (in) (i1)... (in) -1 = i1... i i1,..., ik Xn (i1,..., ik) Sn k m Xn s+i, m = is, s < k;

(m) = i1, m = ik;

m,, m Xn \ {i1,..., ik}.

(i1,..., ik), (j1,..., js) Sn i1,..., ik, j1,..., js Sn = k, 1 k n k0 = k, k1 = k, k2 = 2k,..., kl = lk l+1k = sk 0 s l s = 0.

s > 0 (ks-1) = (kl) X = {1,..., n} ks-1 = kl l {k0, k1,..., kl} k0 k1... kl-1 kl k1 k2... kl k j, 1 j n j {k0, k1,..., kl} / {j0, j1,..., jt} j0 j1... jt-1 jt j1 j2... jt j k0, k1,..., kl, j0, j1,..., jt jr = kq j0 = -rjr {k0, k1,..., kl}, k0 k1... kl-1 kl j0 j1... jt-1 jt = .

k1 k2... kl k0 j1 j2... jt j =.

Sn (i1,..., ik) Sn (i1,..., ik)-1 = ((i1),..., (ik)).

(i1,..., ik) = (i1, i2)(i2, i3) (ik-1, ik).

i1,..., in Xn is, it s < t is > it (-1)M M Sn i1 = 1,..., in = n (-1) ik = k k y1,..., yn yt - ys (-1)sigma =.

yt - ys 1s

- (-1) = (-1) 1... n -1 = =.

1... n -1 -1 = (-1) = (-1) = (-1) (-1).

Sn i1... in 1... n =, =.

1... n 1... n -1 2... 1 2...

(i, j) = (1, 2).

i j... i j...

(i, j) (-1)(1,2) = - s (-1)s k (-1)k- An Sn n! |Sn| = n!, |An| =.

(1, 2) An Sn \ An a11 a1n.

A =.............

an1 ann det A = |A| det A = (-1)a1,1 an,n.

Sn A det A = a11 ann.

A aij = 0 i > j det A (-1)a1,1 an,n Sn 1 1, 2 2,..., n n 1,..., n n = n, (n - 1) = n - 1,..., 1 = A det A s asj = a sj + a sj j = 1,..., n det A = (-1)a1,1 as-1,(s-1)(a s,s + a s,s)as+1,(s+1) an,n = Sn (-1))a1,1 as-1,(s-1)a s,sas+1,(s+1) an,n + Sn (-1)a1,1 as-1,(s-1)a s,sas+1,(s+1) an,n.

Sn A det A = A i j aik = ajk k i < j a1,1 ai,i aj,j an,n (-1) a1,1 ai,j aj,i an,n (-1)(i,j) = -(-1) A Ai Aj i < j Ai A = .

Aj Ai + Aj .

Aj Ai + Aj, Ai Aj = + = det A.

Aj Aj Aj B i j Ai + Aj i j A Ai + Aj B = .

Ai + Aj det B = Ai + Aj Ai Aj Ai Ai Aj Aj Ai Aj = + = + + + = + Ai + Aj Ai + Aj Ai + Aj Ai Aj Ai Aj Aj Ai n = 2 n- x1 1... 0 x2 - x1... xn - xW (x1,..., xn) = 0 x2 - x2x1... x2 - xnx1 = 2 n........................................

0 xn-1 - xn-2x1... xn-1 - xn-2x2 2 n n x2 - x1... xn - xx2 - x2x1... x2 - xnx2 n..................................... = xn-1 - xn-2x1... xn-1 - xn-2x2 2 n n 1 1... x2 x3... xn (x2 - x1) (xn - x1) x2 x2... x2 = 2 3 n........................

xn-2 xn-2... xn-2 3 n (x2 - x1) (xn - x1)W (x2,..., xn) = (xj - xi) = (xj - xi).

2i

-E B t A -E t 0 B det A det B j > n n b1j, b2j, , bnj A AB, -E n AB A (-1)n = det(Ab)(-1)2n = det(AB).

0 -E Mij, 1 i, j n, n - 1 i j Aij (-1)i+jMij A i = 1,..., n det A = ai1Ai1 + + ainAin.

n (ai1,..., ain) = (0,..., 0, aij, 0,..., 0) j= i (0,..., 0, aij, 0,..., 0) (-1)i- (-1)j-1 (-1)i+j aijMij i = j ai1Aj1 + + ainAjn = 0.

A j i A = (aij) Mat(n) A Mat(n) (i, j) Aji A Mat(n) AA = AA = |A|En (i, j) AA ai1Aj1 + + ainAjn = 0.

A Mat(n) A En |A| = A B |A|, |B| |A| A B B = E A Mat(n) A-1 Mat(n) A AA-1 = A-1A = En i = j = 0 (E + Eij)-1 = E - Eij Di()-1 = Di(-1) A- B, C A AC = E = BA B = B(AC) = (BA)C = A Mat(n) A- det A = A- 1 = det A det(A-1) det A = det A = 1 Aji B = = (bij), bij =.

|A| det A BA = AB = E B = A-1.

AX = b A det A = 0 det A = det A i xi =, det A A i A i b AX = b A det A = det A = 0 A det A = 0 A-1 A- X = A-1b A i i A-A-1 AX = En AX = B A Mat(n), det A = 0 X, B Mat(n m) (A|B) (E|C) C = X AX = B Z1 Zk Z1 ZkAX = Z1 ZkB Z1 ZkA = E X = Z1 ZkB = C AX = B X = A-1B V V x + y V x x (x + y) + z = x + (y + z) x, y, z V x + y = y + x x, y V 0 V x + 0 = x x V x V -x V x x + (-x) =, x V ()x = (x), x V ( + )x = x + x x, y V (x + y) = x + y x V 1x = x Mat(n m) R2 R3 [a, b] 0x = 0 = (-1)x = -x 0x = (0 + 0)x = 0x + 0x 0 = 0x + (-0x) = (0x + 0x) + (-0x) = 0x + (0x + (-0x)) = 0x + 0 = 0x.

( - )x = x - x, (x - y) = x - y.

x1,..., xn 1,..., n 1x1 + + nxn = 0.

x1,..., xn 1,..., n 1 = = n = 0.

x x1,..., xn 1,..., n x = 1x1 + + nxn.

x1,..., xn x1,..., xn A A A t det A = det A = t A = n A = 0 A 1,..., n a1,..., an b1,..., bm ai b1,..., bm n > m a1,..., an a1 = b111 + + bmm.....................................

an = b11n + + bmmn 11x1 + + a1nxn =....................................

x1 + + amnxn = m c1,..., cn c1a1 + + cnan = 0.

a1,..., an b1,..., bm ai b1,..., bm a1,..., an n m a1,..., an b1,..., bm, a1,..., an = b1,..., bm.

n = m a1,..., an L L a1,..., an L = a1,..., an.

dim V V V Eij Mat(n m) e = (e1,..., en) V a1,..., ak a1,..., ak, e1 e1 a1,..., an.

e a1,..., ak, ea1,..., an, ei,..., ei, 1 s L e. L = V e = (e1,..., en) V x V x x = x1e1 + + xnen = eX, X = xn x e. X x e.

L e = (e1,..., en) e = (e 1,..., e n) i, j = 1,..., n ei = e 1c1i + + e ncni, e j = e1c 1j + + enc nj.

e = e C, e = eC, C = (cij), C = (c ij) Mat(n) C C e e e e).

e = eCC CC = En C C = En C = C-1 C e = eC e A B =, n A = B = (a1,..., an) = 0.

B r a1,..., ar i = r + 1,..., m b1,i br,i B = 0, - i ai = a1b1,i + + arbr,i.

A A A A A AM M A 1,..., n 1 A = 0, (AM) = 0, n n A AM B r A r r M r A r + A M M r A M i j i < r, j < r 0 = a1jA1,r+1 + + arjAr,r+1 + aijM.

M = A1,r+1 Ar,r+aij = a1j(- ) + + arj(- ).

M M As,r+-, s = 1,..., r, M j j i r A r A AX = AX = n n - r(A) r(A) A A B r = r(A) A i = r + b1,i br,i ei = = 0, - -1 i ei Aei = Bei = a a = an b = a - ar+1er+1 - - anen b = er+1,..., en AX = b A (A|b) A = b A (A|b) A (A|b) A (A|b) b A A, C, D = AC D A D C A r(D) r(C) r(C) = r(A-1D) r(D). r(C) = r(D).

a -b z =, a, b R.

b a C z |z| = det z C z = 0 |z| > z a1 -bz1 = C.

b1 aa + a1 -(b + b1) z + z1 = C b + b1 a + aaa1 - bb1 -(ab1 + ba1) zz1 = C.

ab1 + ba1 aa1 - bb C C z -z z a b a b z-1 = C.

a2 + b2 -b a C R 0 -E = E2, i =.

1 i2 = -E.

a aE.

z z = a + ib z z = a + bi z = a - bi z z z z z1z2 = z1z2. zz = |z|2 z-1 =.

|z2| z = a + bi R a, b |z| z |z1 +z2| |z1|+|z2| |z1 +z2| |z1|-|z2| z1, z2 C z z OX 2n, n Z z = a + bi = x y cos =, sin =.

|z| |z| z = |z|(cos + i sin ) z z1 = |z1|(cos +i sin ).

zz1 = |z||z1|(cos( + ) + i sin( + )).

a R exp(i) = cos + i sin.

z z = |z| exp(i).

z n Z zn = |z|n exp(ni).

z C n Z n z t tn = z n z z t = |t| exp(i).

z = tn = |t|n exp(ni).

n |z| = |t|n |t| = |z| ni (mod 2m), m Z + 2m =, m = 0, 1,..., n - 1.

n + 2m + 2m n t = |z| cos + i sin, m = 0, 1,..., n - 1.

n n z = 0 n n z = 1 m 2m 2m m = cos + i sin.

n n n G xy (xy)z = x(yz) x, y, z G 1 G G x1 = 1x = x x G x G x-x, xx-1 = x-1x = G |G| G x G x-1 x G, (x-1)-1 = x H G H H G G 1 H H G H G x, y H, xy-1 H x, y H x, y-1 H xy-1 H y H y, y H 1 = yy-1 H 1, y H y-1 = 1y-1 H x, y H x, y-1 H x(y-1)-1 = xy H Sn An, Sn- GL(n, C) GL(n, R), GL(n, Q), SL(n, C), SL(n, R);

Un = {z C|zn = 1} i 0 0 -1 0 -i I =, J =, K = SL(2, C).

0 -i 1 0 -i I2 = J2 = K2 = -E, IJ = K, JK = I, KI = J, JI = -K, KJ = -I, IK = -J E, I, J K Q8 SL(2, C).

Hi, i I G, iI Hi G Zn = {0, 1,..., n - 1} Zn n Zn a G n 1, n = 0;

a a, n > 0;

an = n (a-n )-1, n < 0.

a n, m Z an+m = anam, (an)m = anm.

a |a| o(a) a n, an = 1.

n a |a| = n <, m Z n|m (n m) am = a G a {an|n Z} a a G a G a G a G a, a = G Z Un n 2i 2i 2i exp = cos + i sin ;

n n n Zn n a | a | = |a|.

ar = am r < m am-r = 1, |a| = n <.

a = {1, a, a2,..., an-1}.

|a| = n a a n H G = a H = H am, m = m < 0 H a-m, -m > m b = am H ar H, r Z r m r = sm + q, 0 q < m aq = ar-sm = ar(am)-s H, m q > m1,..., mn Z d m1,..., mn u1,..., un Z m1u1 + + mnun = d H = Zm1 + + Zmn H Z H = Zd d = (m1,..., mn) G = a H G n n d n H = ad.

d H = ak 0 k < n d = (n, k) H = ad.

n a G n |ak| =.

(n, k) G = a H = ak n G H = ad d = (n, k) |ak| = |H| =.

d G = a ak G n (k, n) = a f : G H f(xy) = f(x)f(y) x, y G (R, +) (R>0, ) f exp n Un Z G = a f : G Un n 2i f(ak) = exp( ).

n G = a f : G Z f(ak) = k Zn Un H G g G gH {gh|h H} G GL(n, C) SL(n, C) Z nZ Sn Sn- H G x, y G.

xH = yH x-1y H H G x G |H| = |xH| H G x, y G y xH xH = yH yH xH y = xh h H u H xu = y(h-1u), h-1u H xH yH xH = yH H G G H H G |G| = |H|j j G H G H x G xH 0x = x0 = Mat(n, R) Mat(n, k) GL(n, k) Mat(n, k) Zn Zn k Zn (k, n) = k Zn (k, n) > k u Zn ku = 1 Z ku + nv = 1 (k, n) = k Zn (k, n) > d = (k, n) > 1 n > m = nd-1, m = 0 Zn km = k1dm = k1n = 0 Zn Q, R, C Q[i], Q[ 5] char Zn n char Zp = p f : R R f(x + y) = f(x) + f(y), f(xy) = f(x)f(y) R R Mat(n, Mat(m, R)) Mat(nm, R) f : R R f(1) = K R K x, y K x + y, xy K x-1 K, x = Z Q Q R, C Mat(n, Z) Mat(n, R) R x, y R xy = 0, x = 0, y = Z a, b, c a = ab = ac b = c R R[X] a = (a0, a1,... ) R R[X] a b = (b0, b1,... ), c = ab = (c0, c1,... ), k k ck = aibk-i.

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