Linear Algebra - Sebastian Pancratz

v1, . . . , vm w1, . . . , wn m ≤ n vi
wi V n ≤ m wi
vi V
dimF F n = n
dimR P2(R) = 3
dimR C = 2
dimF F = 1
V F v1, . . . , vk
k ≥ 0 v1, . . . , vk, vk+1, . . . , vn
V
v1, . . . , vk V vk+1 ∈ V \ 〈v1, . . . , vk〉
v1, . . . , vk+1 dim V − k
V U ≤ V dim U ≤ dim V
U = V
V F dim V = n
n
n
dimF V = n
v1, . . . , vn
v1, . . . , vn
v1, . . . , vn V
S ⊂ V U V
S U = 〈S〉 S U
S
U S U
V U
S
V = R R S = {1, x, x 2 , . . . } 〈S〉 = P (R)
U, W ≤ V U + W = {u + w : u ∈ U, w ∈ W } U + W ≤ V