Linear Algebra

34 Chapter 1. Linear Systems
position, as a column. For instance, the ‘one over and two up’ vectors above are
denoted in this way.
� �
1
2
More generally, the plane vector starting at (a1,a2) and stretching to (b1,b2) is
denoted
� �
b1 − a1
b2 − a2
since the prior paragraph shows that when the vector starts at the origin, it
ends at this location.
We often just say “the point
� �
1
”
2
rather than “the endpoint of the canonical position of” that vector. That is, we
shall find it convienent to blur the distinction between a point in space and the
vector that, if it starts at the origin, ends at that point. Thus, we will refer to
both of these as Rn .
{(x1,x2) � � �
x1 ��
� x1,x2 ∈ R} { x1,x2 ∈ R}
In the prior section we defined vectors and vector operations with an algebraic
motivation;
� � � � � � � � � �
v1 rv1 v1 w1 v1 + w1
r · =
+ =
v2 rv2 v2 w2 v2 + w2
we can now interpret those operations geometrically. For instance, if �v represents
a displacement then 3�v represents a displacement in the same direction
but three times as far, and −1�v represents a displacement of the same distance
as �v but in the opposite direction.
−�v
�v
And, where �v and �w represent displacements, �v + �w represents those displacements
combined.
�v + �w
�v
3�v
�w
x2