# Linear Algebra/Vectors in Space

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\"Higher-dimensional geometry\" sounds exotic. It is exotic---
interesting and eye-opening. But it isn\'t distant or unreachable.

We begin by defining one-dimensional space to be the set $\mathbb{R}^1$.
To see that definition is reasonable, draw a one-dimensional space

![](Linalg_line.png "Linalg_line.png"){width="800"}

and make the usual correspondence with $\mathbb{R}$: pick a point to
label $0$ and another to label $1$.

![](Linalg_line_with_unit.png "Linalg_line_with_unit.png"){width="800"}

Now, with a scale and a direction, finding the point corresponding to,
say $+2.17$, is easy--- start at $0$ and head in the direction of $1$
(i.e., the positive direction), but don\'t stop there, go $2.17$ times
as far.

The basic idea here, combining magnitude with direction, is the key to
extending to higher dimensions.

An object comprised of a magnitude and a direction is a **vector** (we
will use the same word as in the previous 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 somewhere.

![](Linalg_vectors_1.png "Linalg_vectors_1.png")

There is a subtlety here--- these vectors

![](Linalg_vectors_2.png "Linalg_vectors_2.png"){width="150"}

are equal, even though they start in different places, 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 (\"vector\" is Latin for \"carrier\" or
\"traveler\"). These squares undergo the same displacement, despite that
those displacements start in different places.

![](Linalg_vectors_3.png "Linalg_vectors_3.png")

Sometimes, to emphasize this property vectors have of not being
anchored, they are referred to as **free** vectors. Thus, these free
vectors are equal as each is a displacement of one over and two up.

![](Linalg_vectors_4.png "Linalg_vectors_4.png")

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 $b_2-a_2=d_2-c_2$.

An expression like \"the vector that, were it to start at $(a_1,a_2)$,
would extend to $(b_1,b_2)$\" is awkward. We instead describe such a
vector as

$$\begin{pmatrix} b_1-a_1 \\ b_2-a_2 \end{pmatrix}$$

so that, for instance, the \"one over and two up\" arrows shown above
picture this vector.

$$\begin{pmatrix} 1 \\ 2 \end{pmatrix}$$

We often draw the arrow as starting at the origin, and we then say it is
in the **canonical position** (or **natural position**). When the vector

$$\begin{pmatrix} b_1-a_1 \\ b_2-a_2 \end{pmatrix}$$

is in its canonical position then it extends to the endpoint
$(b_1-a_1,b_2-a_2)$.

We typically just refer to \"the point

$$\begin{pmatrix} 1 \\ 2 \end{pmatrix}$$\"

rather than \"the endpoint of the canonical position of\" that vector.

Thus, we will call both of these sets $\mathbb{R}^2$.

$$\{(x_1,x_2)\,\big|\, x_1,x_2\in\mathbb{R}\}
\qquad
\{\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}\,\big|\, x_1,x_2\in\mathbb{R}\}$$

In the prior section we defined vectors and vector operations with an
algebraic motivation;

$$r\cdot\begin{pmatrix} v_1 \\ v_2 \end{pmatrix}
=
\begin{pmatrix} rv_1 \\ rv_2 \end{pmatrix}
\qquad
\begin{pmatrix} v_1 \\  v_2 \end{pmatrix}
+
\begin{pmatrix} w_1 \\ w_2 \end{pmatrix}
=
\begin{pmatrix} v_1+w_1 \\ v_2+w_2 \end{pmatrix}$$

we can now interpret those operations geometrically. For instance, if
$\vec{v}$ represents a displacement then $3\vec{v}\,$ represents a
displacement in the same direction but three times as far, and
$-1\vec{v}\,$ represents a displacement of the same distance as
$\vec{v}\,$ but in the opposite direction.

![](Linalg_scaled_vectors.png "Linalg_scaled_vectors.png")

And, where $\vec{v}$ and $\vec{w}$ represent displacements,
$\vec{v}+\vec{w}$ represents those displacements combined.

![](Linalg_vector_addition.png "Linalg_vector_addition.png")

The long arrow is the combined displacement in this sense: if, in one
minute, a ship\'s motion gives it the displacement relative to the earth
of $\vec{v}$ and a passenger\'s motion gives a displacement relative to
the ship\'s deck of $\vec{w}$, then $\vec{v}+\vec{w}$ is the
displacement of the passenger relative to the earth.

Another way to understand the vector sum is with the **parallelogram
rule**. Draw the parallelogram formed by the vectors
$\vec{v}_1,\vec{v}_2$ and then the sum $\vec{v}_1+\vec{v}_2$ extends
along the diagonal to the far corner.

![](Linalg_vector_addition_2.png "Linalg_vector_addition_2.png")

The above drawings show how vectors and vector operations behave in
$\mathbb{R}^2$. We can extend to $\mathbb{R}^3$, or to even
higher-dimensional spaces where we have no pictures, with the obvious
generalization: the free vector that, if it starts at
$(a_1,\ldots,a_n)$, ends at $(b_1,\ldots,b_n)$, is represented by this
column

$$\begin{pmatrix} b_1-a_1 \\ \vdots \\ b_n-a_n \end{pmatrix}$$

(vectors are equal if they have the same representation), we aren\'t too
careful to distinguish between a point and the vector whose canonical
representation ends at that point,

$$\mathbb{R}^n=
\{\begin{pmatrix} v_1 \\ \vdots \\ v_n \end{pmatrix}\,\big|\, v_1,\ldots,v_n\in\mathbb{R}\}$$

and addition and scalar multiplication are component-wise.

Having considered points, we now turn to the lines.

In $\mathbb{R}^2$, the line through $(1,2)$ and $(3,1)$ s comprised of
(the endpoints of) the vectors in this set

$$\{ \begin{pmatrix} 1 \\ 2 \end{pmatrix}+t\cdot\begin{pmatrix} 2 \\ -1 \end{pmatrix}\,\big|\, t\in\mathbb{R}\}$$

That description expresses this picture.

![](Linalg_vector_subtraction.png "Linalg_vector_subtraction.png")

The vector associated with the parameter $t$ has its whole body in the
line--- it is a **direction vector** for the line. Note that points on
the line to the left of $x=1$ are described using negative values of
$t$.

In $\mathbb{R}^3$, the line through $(1,2,1)$ and $(2,3,2)$ is the set
of (endpoints of) vectors of this form

![](Linalg_line_in_R3.png "Linalg_line_in_R3.png")

and lines in even higher-dimensional spaces work in the same way.

If a line uses one parameter, so that there is freedom to move back and
forth in one dimension, then a plane must involve two. For example, the
plane through the points $(1,0,5)$, $(2,1,-3)$, and $(-2,4,0.5)$
consists of (endpoints of) the vectors in

$$\{ \begin{pmatrix} 1 \\ 0 \\ 5 \end{pmatrix}
+t\cdot\begin{pmatrix} 1 \\ 1 \\ -8 \end{pmatrix}
+s\cdot\begin{pmatrix} -3 \\ 4 \\ -4.5 \end{pmatrix}
\,\big|\, t,s\in\mathbb{R}      \}$$

(the column vectors associated with the parameters

$$\begin{pmatrix} 1 \\ 1 \\ -8 \end{pmatrix}
=
\begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}
-
\begin{pmatrix} 1 \\ 0 \\ 5 \end{pmatrix}
\qquad
\begin{pmatrix} -3 \\ 4 \\ -4.5 \end{pmatrix}
=
\begin{pmatrix} -2 \\ 4 \\ 0.5 \end{pmatrix}
-
\begin{pmatrix} 1 \\ 0 \\ 5 \end{pmatrix}$$

are two vectors whose whole bodies lie in the plane). As with the line,
note that some points in this plane are described with negative $t$\'s
or negative $s$\'s or both.

A description of planes that is often encountered in algebra and
calculus uses a single equation as the condition that describes the
relationship among the first, second, and third coordinates of points in
a plane.

![](Linalg_plane.png "Linalg_plane.png")

The translation from such a description to the vector description that
we favor in this book is to think of the condition as a one-equation
linear system and parametrize $x=(1/2)(4-y-z)$.

![](Linalg_plane_with_basis.png "Linalg_plane_with_basis.png")

Generalizing from lines and planes, we define a **$k$-dimensional linear
surface** (or **$k$-flat**) in $\mathbb{R}^n$ to be
$\{\vec{p}+t_1\vec{v}_1+t_2\vec{v}_2+\cdots+t_k\vec{v}_k \,\big|\, t_1,\ldots ,t_k\in\mathbb{R}\}$
where $\vec{v}_1,\ldots,\vec{v}_k\in\mathbb{R}^n$. For example, in
$\mathbb{R}^4$,

$$\{\begin{pmatrix} 2 \\ \pi \\ 3 \\ -0.5 \end{pmatrix}
+t\begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}
\,\big|\, t\in\mathbb{R}\}$$

is a line,

$$\{
\begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}
+t\begin{pmatrix} 1 \\ 1 \\ 0 \\ -1 \end{pmatrix}
+s\begin{pmatrix} 2 \\ 0 \\ 1 \\ 0 \end{pmatrix}
\,\big|\, t,s\in\mathbb{R}\}$$

is a plane, and

$$\{
\begin{pmatrix} 3 \\ 1 \\ -2 \\ 0.5 \end{pmatrix}
+r\begin{pmatrix} 0 \\ 0 \\ 0 \\ -1 \end{pmatrix}
+s\begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \end{pmatrix}
+t\begin{pmatrix} 2 \\ 0 \\ 1 \\ 0 \end{pmatrix}
\,\big|\, r,s,t\in\mathbb{R}\}$$

is a three-dimensional linear surface. Again, the intuition is that a
line permits motion in one direction, a plane permits motion in
combinations of two directions, etc.

A linear surface description can be misleading about the dimension---
this

$$L=\{
\begin{pmatrix} 1 \\ 0 \\ -1 \\ -2 \end{pmatrix}
+t\begin{pmatrix} 1 \\ 1 \\ 0 \\ -1 \end{pmatrix}
+s\begin{pmatrix} 2 \\ 2 \\ 0 \\ -2 \end{pmatrix}
\,\big|\, t,s\in\mathbb{R}\}$$

is a **degenerate** plane because it is actually a line.

$$L=\{
\begin{pmatrix} 1 \\ 0 \\ -1 \\ -2 \end{pmatrix}
+r\begin{pmatrix} 1 \\ 1 \\ 0 \\ -1 \end{pmatrix}
\,\big|\, r\in\mathbb{R}\}$$

We shall see in the Linear Independence section of Chapter Two what
relationships among vectors causes the linear surface they generate to
be degenerate.

We finish this subsection by restating our conclusions from the first
section in geometric terms. First, the solution set of a linear system
with $n$ unknowns is a linear surface in $\mathbb{R}^n$. Specifically,
it is a $k$-dimensional linear surface, where $k$ is the number of free
variables in an echelon form version of the system. Second, the solution
set of a homogeneous linear system is a linear surface passing through
the origin. Finally, we can view the general solution set of any linear
system as being the solution set of its associated homogeneous system
offset from the origin by a vector, namely by any particular solution.

## Exercises


## References

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