Linear Algebra and Geometry
We construct a dictionary between some geometrical notions and some notions from
linear algebra.
Adding, Scalar Multiplication
An element of (
x, y
)
∈
R
2
corresponds to an arrow with tail at the origin in
R
2
and head
at the point (
x, y
).
Two arrows are the same vector if they have the same length and
direction. This works the same way in
R
3
or
R
n
for any
n
.
We can add vectors algebraically. We have (
x
1
, y
1
) + (
x
2
, y
2
) = (
x
1
+
x
2
, y
1
+
y
2
). This
corresponds to the geometrical operation of placing the vector (
x
2
, y
2
) so that its tail is at
the head of (
x
1
, y
1
). The result of adding is the vector with tail at the origin and head at
the head of (
x
2
, y
2
). We can interchange the roles of the two vectors and we obtain the
same result. Addition works the same in
R
3
.
We can multiply a vector (
x, y
) by a scalar
a
∈
R
.
Algebraically we get
a
(
x, y
) =
(
ax, ay
). Geometrically this is the operation of stretching or squeezing the vector (
x, y
) by
a factor of
a
if
a
≥
0. If
a <
0, then we turn the vector around and stretch by a factor of

a

.
We can combine the two operations.
Let
~u
=
1
2
and
~v
=
3
1
.
What is the
geometrical picture of 2
~u

v
?
1
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How do we visualize this? Start by sketching examples such as
2
1
2
+ 3

3
1
,
or

2
1
2
+ 2

3
1
.
The Geometry of Solving Equations in the Plane
We can draw a picture of solving linear equations. We can look at each of the equations
x

3
y
= 1
2
x
+
y
=

5
as defining a line, so the solution to the set of equations is the intersection of the lines.
A single linear equation in variables
x, y
can be thought of as eliminating all the points
in the plane except those satisfying the equation. This description allows us to see what
happens when we are given two linear equations in two unknowns. Three different things
can happen.
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 Fall '08
 MARKMAN
 Linear Algebra, Algebra, Geometry, Multiplication, Scalar, Vector Space, Elementary algebra

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