a11 x1 +... + a1n xn = b1
a21x1 +... + a2nxn = b2
........................ ()
am1x1 +... + amn xn = bm,
. () n , , n- . Rn. .
, m £ n. , m > n. k- a k . , Rn, , a 1,..., a m . ,
a k = a 1 a 1 +... + a k - 1 a k - 1,
a k - a 1 a 1 -... - a k - 1 a k - 1 = 0.
, k- 1-, 2-,..., (k-1)- , a1,..., a k - 1, 0 = b. b ¹ 0, . b = 0, k- . . , , .
a11 x1 +... + a1n xn = 0
a21x1 +... + a2nxn = 0
........................ ()
am1x1 +... + amn xn = 0
.
A x = 0.
,
0 =
i j . . .
. Rn.
. x y .
A(a x + b y) = aA x + bA y = a× 0 + b× 0 = 0.
, () ().
n, n. , (., [ ]).
. n n-r, r - .
, m < n , . . r £ m < n (n-r > 0). ,
a1 x1 +... + an xn = 0
n-1 x = (x1,..., xn) = (1,..., n). . , n = 3,
|
|
a1xx + a2x2 + a3x3 = 0
, = (1, 2, 3).
X0, x + X0 - x + x0, x0 Î X0.
(),
A x = b.
.
. x (). x + X0.
. y (). x0 = x - y ().
A x0 = A(x - y) = b - b = 0.
, y = x0 + x.
, y = x0 + x (), ..
A y = A(x0 + x) = A x0 + A x = 0 + b = b.
() , . , m . .