A First Course in Linear Algebra, 2017a

148 R n
Theorem 4.6: Properties of Scalar Multiplication
The following properties hold for vectors ⃗u,⃗v ∈ R n and k, p scalars.
• The Distributive Law over Vector Addition
k (⃗u +⃗v)=k⃗u + k⃗v
• The Distributive Law over Scalar Addition
(k + p)⃗u = k⃗u + p⃗u
• The Associative Law for Scalar Multiplication
k (p⃗u)=(kp)⃗u
• Rule for Multiplication by 1
1⃗u = ⃗u
Proof. We will show the proof of:
Note that:
k (⃗u +⃗v)
k (⃗u +⃗v)=k⃗u + k⃗v
=k [u 1 + v 1 ···u n + v n ] T
=[k (u 1 + v 1 )···k (u n + v n )] T
=[ku 1 + kv 1 ···ku n + kv n ] T
=[ku 1 ···ku n ] T +[kv 1 ···kv n ] T
= k⃗u + k⃗v
♠
We now present a useful notion you may have seen earlier combining vector addition and scalar multiplication
Definition 4.7: Linear Combination
A vector ⃗v is said to be a linear combination of the vectors ⃗u 1 ,···,⃗u n if there exist scalars,
a 1 ,···,a n such that
⃗v = a 1 ⃗u 1 + ···+ a n ⃗u n
For example,
⎡
3⎣
−4
1
0
⎤
⎡
⎦ + 2⎣
−3
0
1
⎤
⎡
⎦ = ⎣
−18
3
2
⎤
⎦.
Thus we can say that
⎡
⃗v = ⎣
−18
3
2
⎤
⎦