Linear Algebra, 2020a

38 Chapter One. Linear Systems
The long arrow is the combined displacement in this sense: imagine that you are
walking on a ship’s deck. Suppose that in one minute the ship’s motion gives it
a displacement relative to the sea of ⃗v, and in the same minute your walking
gives you a displacement relative to the ship’s deck of ⃗w. Then ⃗v + ⃗w is your
displacement relative to the sea.
Another way to understand the vector sum is with the parallelogram rule.
Draw the parallelogram formed by the vectors ⃗v and ⃗w. Then the sum ⃗v + ⃗w
extends along the diagonal to the far corner.
⃗w
⃗v + ⃗w
The above drawings show how vectors and vector operations behave in R 2 .
We can extend to R 3 , or to even higher-dimensional spaces where we have no
pictures, with the obvious generalization: the free vector that, if it starts at
(a 1 ,...,a n ), ends at (b 1 ,...,b n ), is represented by this column.
⎛ ⎞
b 1 − a 1
⎜
⎝
⎟
. ⎠
b n − a n
Vectors are equal if they have the same representation. We aren’t too careful
about distinguishing between a point and the vector whose canonical representation
ends at that point.
⎛
⎞
v 1
R n ⎜ .
= { ⎝
.
v n
⃗v
⎟
⎠ | v 1 ,...,v n ∈ R}
And, we do addition and scalar multiplication component-wise.
Having considered points, we next turn to lines. In R 2 , the line through
(1, 2) and (3, 1) is comprised of (the endpoints of) the vectors in this set.
( ( 1 2
{ + t | t ∈ R}
2)
−1)
In the description the vector that is associated with the parameter t
( ) ( ) ( )
2 3 1
= −
−1 1 2