De▯nition: Let v ; 1 ; 2▯▯ ; v be vkctors in R . The span of
these vectors is the set of all linear combinations of these vectors and
is denoted by sp(v 1 v ;2▯▯▯ ; v )k
sp(v 1 v 2 ▯▯▯ ; v )k= f▯ v 1 ▯1v + 2▯▯2+ ▯ v j ▯ ; ▯k▯ k▯ 2 1g. k
De▯nition: Let v = [v 1v ;2▯▯▯ ;v ]nbe a vector in R . The norm
(or magnitude or length) of v is kvk = v + v + ▯▯▯ + v . 2
1 2 n
Theorem (Properties of Norm)
For all vectors v; w 2 R and for all scalars ▯, we have
(1) kvk ▯ 0 and kvk = 0 if and only if v = 0. (positivity)
(2) k▯vk = j▯jkvk (homogeneity)
(3) kv + wk ▯ kvk + kwk (triangle inequality) n
De▯nition: A vector in R , with norm 1, is called a unit vector.
If v 6= 0 is a vector in R , a unit vector with the same direction as
v is v.
Note: The standard basis vectors are unit vectors and are sometimes
called unit coordinate vectors.