Linear Algebra, 2020a

36 Chapter One. Linear Systems
to find the point matching +2.17, start at 0 and head in the direction of 1, and
go 2.17 times as far.
The basic idea here, combining magnitude with direction, is the key to
extending to higher dimensions.
An object in an R n that is comprised of a magnitude and a direction is a
vector (we use the same word as in the prior section because we shall show
below how to describe such an object with a column vector). We can draw a
vector as having some length and pointing in some direction.
There is a subtlety involved in the definition of a vector as consisting of a
magnitude and a direction — these
are equal, even though they start in different places They are equal because they
have equal lengths and equal directions. Again: those vectors are not just alike,
they are equal.
How can things that are in different places be equal? Think of a vector as
representing a displacement (the word ‘vector’ is Latin for “carrier” or “traveler”).
These two squares undergo displacements that are equal despite that they start
in different places.
When we want to emphasize this property vectors have of not being anchored
we refer to them as free vectors. Thus, these free vectors are equal, as each is a
displacement of one over and two up.
More generally, vectors in the plane are the same if and only if they have the
same change in first components and the same change in second components: the
vector extending from (a 1 ,a 2 ) to (b 1 ,b 2 ) equals the vector from (c 1 ,c 2 ) to
(d 1 ,d 2 ) if and only if b 1 − a 1 = d 1 − c 1 and b 2 − a 2 = d 2 − c 2 .
Saying ‘the vector that, were it to start at (a 1 ,a 2 ), would extend to (b 1 ,b 2 )’
would be unwieldy. We instead describe that vector as
(
b 1 − a 1
b 2 − a 2
)