Vector addition and scalar multiplication
Using components
• Let u =< u1 , u2, ..., un > and v =< v1 , v2, ..., vn > be vectors and let c be a scalar. We define
vector addition:
u + v =< u1 + v1 , u2 + v2, ..., un + vn >
and scalar multiplication:
cv =< cv1 , cv2, ..., cvn >
Example: For u =< 1, 4, 5 >, v =< −1, 5, 2 >, c = −4, find
cu + v
• Properties of vector addition:
* u + v = v + u (commutative property of addition)
* u + (v + w) = (u + v) + w (associative property of addition)
* u + 0 = 0 + u = u (additive identity; the identity element is the zero vector, a vector with
ui = 0 for all i)
* u + −u = −u + u = 0 (additive inverse)
• Properties of scalar multiplication:
* c(u + v) = cu + cv (distributive property; scalar over vector sum)
* (c + d)u = cu + du (distributive property; vector over scalar sum)
* c(du) = (cd)u (associative; scalars with vector)
* 1u = u (multiplicative identity for scalar multiplication)
Proof: Prove that for vectors u and v,
u+v =v+u
Suppose that we have two vectors in n dimensional space:
u = < u1 , u2, ..., un >
v = < v1 , v2, ..., vn >
By the definition of vector addition,
u + v =< u1 + v1 , u2 + v2, ..., un + vn >
Since the components themselves are scalars, by the commutative property of addition
u + v = < u1 + v1, u2 + v2, ..., un + vn >
= < v1 + u1, +v2 + u2, ..., vn + un >
= v+u
• The set of all vectors in the plane (v =< v1, v2 >) with accompanying scalars and the operations
of vector addition and scalar multiplication defined on them form a vector space, R2 . In general,
for any n, the set of all vectors in the form v =< v1, v2, ..., vn > with accompanying scalars form a
vector space Rn under the operations of vector addition and scalar multiplication.