(V, K, •", ") K

(V, •", ")




Ä…1, Ä…2, . . . , Ä…n " K v1, v2, . . . , vn " V

(Ä…1 " v1) •" (Ä…2 " v2) •" · · · •" (Ä…n " vn)

v1, v2, . . . , vn Ä…1, Ä…2, . . . , Ä…n
n

(Ä…1 " v1) •" (Ä…2 " v2) •" · · · •" (Ä…n " vn) a" Ä…i " vi
i=1


v1, v2, . . . , vn " V

n
"Ä…1, Ä…2, . . . , Ä…n " K Ä…i " vi = 0 =Ò! Ä…1 = Ä…2 = · · · = Ä…n = 0
i=1

0 V Rn

(0, . . . , 0)

¸ 0

K






"Ä…1, Ä…2, . . . , Ä…n " R Ä…1v1 + Ä…2v2 + · · · + Ä…nvn = 0 =Ò! Ä…1 = Ä…2 = · · · = Ä…n = 0

V








V

{v1, v2, v3, . . . } = {vi}"
i=1









v1, v2, . . . , vn " V




Rn

v1, v2, . . . , vn " Rn

vi









v1, v2, . . . , vn " V V

v " V Ä…1, Ä…2, . . . , Ä…n " R
v = Ä…1v1 + Ä…2v2 + · · · + Ä…nvn



{v1, v2, . . . , vn} ‚" V V



V



V



v " V {v1, v2, . . . , vn} v = Ä…1v1 +


Ä…2v2+· · ·+Ä…nvn Ä…1, Ä…2, . . . , Ä…n v

{vi}i=1,...,n



Rn












V





u1 = 2, u2 = -3
w1 = (3, 1), w2 = (-2, 1)

v1 = (-1, 2), v2 = (2, 6), v3 = (3, 2)
z1 = (1, 1, -2), z2 = (3, 1, 0), z3 = (-1, 2, 1)




(R2, R, •", ")
(x1, x2) •" (y1, y2) := (x1 + y1 + 1, x2 + y2)
Ä… " (x1, x2) := (Ä…x1 + Ä… - 1, Ä…x2)



(2, 1) (4, 2)

(2, 1) (5, 2)



R3

z1 = (1, 1, -2) z2 = (3, 1, 0) z3 = (-1, 2, 1)


R3 z = (1, -3, -3)





R4

v1 = (1, 2, 1, -2) v2 = (2, 3, 0, -1) v3 = (1, 2, 1, 3) v4 = (1, 3, -1, 0)




v = (7, 14, -1, 1)



{v1, v2, v3} R3

{v1, v2, v3} v1 = v1 - v2 v2 = v1 + v2 + v3


v3 = 2v1 - v2 - v3

v = 3v1 + 4v2 + 3v3






R2[x]

u1(x) = 1 + x u2(x) = 1 - x
u3(x) = 1 + x + x2




u(x) = x2 + 2x + 1






v1 = (1, 1, 1), v2 = (1, 1, 0), v3 = (1, 0, 0)
v1 = (0, 1, 0, 1), v2 = (0, 3, 0, -1), v3 = (3, 2, 1, 0), v4 = (0, -3, 1, 2)





V = {(x - 5y, x + y, 2x + y, x + y) " R4 : x, y " R}

V

R4





v = (-2, 4, 7, 4)