CHAPTER I: LINEAR ALGEBRA - OCW UPM

Examples of vector spaces
AFFINE AND PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda
1. R n = {(a 1 , a 2 , . . . , a n )|a i ∈ R, i = 1, . . . n}, n ∈ N is a real vector space
with the operations:
(a 1 , a 2 , . . . , a n ) + (b 1 , b 2 , . . . , b n ) = (a 1 + b 1 , a 2 + b 2 , . . . , a n + b n )
α(a 1 , a 2 , . . . , a n ) = (αa 1 , αa 2 , . . . , αa n )
con (b 1 , b 2 , . . . , b n ) ∈ R n y α ∈ R. The zero vector is 0 = (0, 0, . . . , 0). The
opposite vector of (a 1 , a 2 , . . . , a n ) is (−a 1 , −a 2 , . . . , −a n ).
2. Given a homogeneous system of linear equations with real coefficients
⎧
a 11 x 1 + . . . + a 1n x n = 0
⎪⎨
a
(∗) 21 x 1 + . . . + a 2n x n = 0
· · ·
⎪⎩ a m1 x 1 + . . . + a mn x n = 0.
with n unknowns. The set of solutions of the system (∗):
W = {(a 1 , a 2 , . . . , a n ) ∈ R n |(a 1 , a 2 , . . . , a n ) is a solution of (∗)} ⊆ R n
is a real vector space.