Definition 3.2.1.
Consider any set of equations in the variables \(x_1, x_2, \ldots, x_n\text{.}\) The locus in \(\RR^n\) of this set of equations is the set of vectors that satisfy all of the equations. The plural of a locus is loci.
Section 3.2 Loci
Subsection 3.2.1 Definition of a Locus
Spans were the first way to define linear and affine subspaces. There is another approach to describe these object, which I describe in this section. I’ll start with the main definition.
In general, the equations can be of any sort. The unit circle in \(\RR^2\) is most commonly defined as the locus of the equation \(x^2 + y^2 = 1\text{.}\) The graph of a function is the locus of the equation \(y = f(x)\text{.}\) However, linear algebra excludes curved objects. The subset is concerned with linear/affine objects: things that are straight and flat. So, I don’t want loci of just any kind of equation: I want loci of linear equations. The idea is likely familiar, but let me formally define it anyway.
Definition 3.2.2.
Let \(a_i\) and \(c\) be real numbers. A linear equation in the variables \(x_1, x_2, \ldots x_n\) is an equation of the following form.
\begin{equation*}
a_1 x_1 + a_2 x_2 + \ldots + a_n x_n = c
\end{equation*}
Proposition 3.1.6 claimed that all linear subspaces were described by spans. Shortly afterward, I showed that all affine subspaces were described by offset spans. In addition to spans, I can also use loci to describe any affine or linear subspace.
Proposition 3.2.3.
Any linear or affine subspace of \(\RR^n\) can be described as the locus of finitely many linear equations. Likewise, the locus of any number of linear equations is either an affine subspace of \(\RR^n\) or the empty set.
It is quite valuable to have these two different descriptions of linear and affine subspaces. Each description (spans or loci) has strengths and weaknesses. Depending on what calculations I want to do, using either a span or a locus description may help a great deal.
There is also a nice contrast in the constructions of spans and loci. Spans are essentially built up by adding more and more vectors, more and more directions. The span of no vectors is the origin. The span of one vector is a line. The span of two linearly independent vectors is a plane. This process continues (if the ambient dimension is large enough) to make object of higher and higher dimensions as more linearly independent vectors are added to the span.
To be more specific, in \(\RR^2\text{,}\) adding the equation \(x=3\) restricts to a vertical line passing through the \(x\)-axis at \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\text{.}\) Likewise, the equation \(y=4\) restricts to a horizontal line passing through the \(y\)-axis at \(\begin{pmatrix} 0 \\ 4
\end{pmatrix}\text{.}\) If I consider the locus of both equations, there is only one point remaining: \(\begin{pmatrix}
3 \\ 4 \end{pmatrix}\) is the only point that satisfies both equations. In this way, each additional equation potentially adds an additional restriction and leads to a smaller linear or affine subspaces.
As with spans, there can be the problem of redundant information. In \(\RR^3\text{,}\) the locus of the equation \(x + y
+ z = 1\) is a plane. If I also add the equation \(2x + 2y +
2z = 2\text{,}\) however, I don’t get a line. The second equation is a multiple of the first, and any points that satisfy the first equation also satisfy the second. The second equation is redundant and unnecessary. Determining what is redundant (and determining the dimension of a locus) will rely on calculation tools introduced in Section 5.3.
Subsection 3.2.2 Familiar Loci
I can define some familiar objects which are presented as the locus of one linear equation.
Definition 3.2.4.
A line in \(\RR^2\) is the locus of the equation \(ax + by = c\) for \(a,b,c \in \RR\text{.}\) In general, the line is affine. The line is linear if \(c=0\text{.}\)
Definition 3.2.5.
A plane in \(\RR^3\) is the locus of the linear equation \(ax + by + cz = d\text{.}\) In general, the plane is affine. The plane is linear if \(d=0\text{.}\)
If we think of a plane in \(\RR^3\) as the locus of one linear equation, the important dimensional fact about a plane is not that it has dimension two but that it has dimension one less than its ambient space \(\RR^3\text{.}\)
Definition 3.2.6.
A hyperplane in \(\RR^n\) is the locus of one linear equation: \(a_1 x_1 + a_2 x_2 + \ldots + a_n x_n
= c\text{.}\) It has dimension \(n-1\text{.}\) It is, in general, affine. The hyperplane is linear if \(c=0\text{.}\)
Subsection 3.2.3 Intersection
Definition 3.2.7.
If \(A\) and \(B\) are sets, their intersection \(A
\cap B\) is the set of all points they have in common. The intersection of affine subspaces is also an affine subspace. If \(A\) and \(B\) are both linear, the intersection is also linear.
Example 3.2.8.
Loci of more than one equation can be understood as intersections. Consider the locus of two equations, say the example I used from \(\RR^2\) before: the locus of \(x=3\) and \(y=4\text{.}\) I defined this directly as a single locus. However, I could just as easily think of this as the intersection of the two lines given by \(x=3\) and \(y=4\) separately. In this way, it is the intersection of two loci. Similarly, all loci are the intersection of the planes or hyperplanes defined by each individual linear equation.
