A First Course in Linear Algebra, 2017a

146 R n
components are equal. Precisely,
⃗u =⃗v if and only if
u j = v j for all j = 1,···,n
Thus [ 1 2 4 ] T ∈ R 3 and [ 2 1 4 ] T ∈ R 3 but [ 1 2 4 ] T [ ] T ≠ 2 1 4 because, even though
the same numbers are involved, the order of the numbers is different.
For the specific case of R 3 , there are three special vectors which we often use. They are given by
⃗i = [ 1 0 0 ] T
⃗j = [ 0 1 0 ] T
⃗k = [ 0 0 1 ] T
We can write any vector ⃗u = [ u 1 u 2 u 3
] T as a linear combination of these vectors, written as ⃗u =
u 1
⃗i + u 2
⃗j + u 3
⃗ k. This notation will be used throughout this chapter.
4.2 Algebra in R n
Outcomes
A. Understand vector addition and scalar multiplication, algebraically.
B. Introduce the notion of linear combination of vectors.
Addition and scalar multiplication are two important algebraic operations done with vectors. Notice
that these operations apply to vectors in R n , for any value of n. We will explore these operations in more
detail in the following sections.
4.2.1 Addition of Vectors in R n
Addition of vectors in R n is defined as follows.
Definition 4.3: Addition of Vectors in R n
⎡ ⎤ ⎡ ⎤
u 1 v 1
⎢
If ⃗u = . ⎥ ⎢
⎣ .. ⎦, ⃗v = . ⎥
⎣ .. ⎦ ∈ R n then ⃗u +⃗v ∈ R n andisdefinedby
u n v n
⃗u +⃗v =
=
⎡
⎢
⎣
⎡
⎢
⎣
⎤
u 1
.. . ⎥
⎦ +
u n
⎡
⎢
⎣
⎤
u 1 + v 1
⎥ ... ⎦
u n + v n
⎤
v 1
.. . ⎥
⎦
v n