Chapter 8 (The Geometry of Vector Spaces): Affine combinations (仿射组合)

• This chapter begins with geometric ideas that are already familiar to you. Affine combinations of vectors generalize the concepts of lines and planes to higher dimensions and describe the mathematical objects formed when subspaces are shifted away from the origin. Hyperplanes are a generalization of planes, and polytopes are a generalization of polygons.
• The geometric objects studied in this chapter include line segments, solid triangles and polygons, specials types of curves and surfaces, and generalizations to sets in more than three dimensions. These objects form the building blocks for applications to computer graphics and to linear programming.

Affine combinations

• Given vectors (or “points”) ${\mathbf{v}}_1,{\mathbf{v}}_2,...,{\mathbf{v}}_p$ in ${\mathbb{R}}^n$ and scalars $c_1,...,c_p$, an affine combination of ${\mathbf{v}}_1,{\mathbf{v}}_2,...,{\mathbf{v}}_p$ is a linear combination
  such that the weights satisfy $c_1+...+c_p=1$.

Affine hull / Affine span (仿射包 / 仿射生成集)

Definition

• The affine hull of a single point ${\mathbf{v}}_1$ is just the set $\{{\mathbf{v}}_1\}$. The affine hull of two distinct points is often written in a special way. Suppose $\mathbf{y}=c_1{\mathbf{v}}_1+c_2{\mathbf{v}}_2$ with $c_1+c_2=1$. Write $t$ in place of $c_2$, so that $c_1=1-t$ . Then the affine hull of $\{{\mathbf{v}}_1,{\mathbf{v}}_2\}$ is the set
  This set of points includes ${\mathbf{v}}_1$ (when $t=0$) and ${\mathbf{v}}_2$ (when $t=1$). If ${\mathbf{v}}_2={\mathbf{v}}_1$, then (1) again describes just one point. Otherwise, (1) describes the line through ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$. To see this, rewrite (1) in the form
  where $\mathbf{p}$ is ${\mathbf{v}}_1$ and $\mathbf{u}$ is ${\mathbf{v}}_2-{\mathbf{v}}_1$.
  
  Figure 2 uses the original points ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$, and displays $\text{aff }\{{\mathbf{v}}_1,{\mathbf{v}}_2\}$ as the line through ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$.
  

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• The next example takes a fresh look at a familiar set—the set of all solutions of a system $A\mathbf{x}=\mathbf{b}$.

EXAMPLE 3

• Suppose that the solutions of an equation $A\mathbf{x}=\mathbf{b}$ are all of the form $\mathbf{x}=x_3\mathbf{u}+\mathbf{p}$, (通解 + 特解 的形式)
  Find points ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$ such that the solution set of $A\mathbf{x}=\mathbf{b}$ is $\text{aff }\{{\mathbf{v}}_1,{\mathbf{v}}_2\}$.

SOLUTION

• The solution set is a line through $\mathbf{p}$ in the direction of $\mathbf{u}$. Since $\text{aff }\{{\mathbf{v}}_1,{\mathbf{v}}_2\}$ is a line through ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$, identify two points on the line $\mathbf{x}=x_3\mathbf{u}+\mathbf{p}$. Two simple choices appear when $x_3=0$ and $x_3=1$. That is, take ${\mathbf{v}}_1=\mathbf{p}$ and ${\mathbf{v}}_2=\mathbf{u}+\mathbf{p}$, so that
  
• In this case, the solution set is described as the set of all affine combinations of the form
  

Decide whether a point is in an affine hull

In the statement of Theorem 1, the point ${\mathbf{v}}_1$ could be replaced by any of the other points in the list ${\mathbf{v}}_1,...,{\mathbf{v}}_p$: Only the notation in the proof would change.

PROOF

• [Hint: $\mathbf{y}=(1-c_2-...-c_p){\mathbf{v}}_1+c_2{\mathbf{v}}_2+...+c_p{\mathbf{v}}_p$]

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• We can judge whether an arbitrary vector $\mathbf{y}$ is an affine combination of $\{{\mathbf{v}}_1,...,{\mathbf{v}}_n\}$ by using Theorem 1. But the question can be answered more directly if the chosen points ${\mathbf{v}}_i$ are a basis for ${\mathbb{R}}^n$. For example, let $\mathcal{B}=\{{\mathbf{b}}_1,...,{\mathbf{b}}_n\}$ be such a basis. Then any $\mathbf{y}$ in ${\mathbb{R}}^n$ is a unique linear combination of ${\mathbf{b}}_1,...,{\mathbf{b}}_n$. This combination is an affine combination of the $\mathbf{b}$’s if and only if the weights sum to 1.

Affine set

集合 $S$ 是仿射的

• Geometrically, a set is affine if whenever two points are in the set, the entire line through these points is in the set. (If $S$ contains only one point, $\mathbf{p}$, then the line through $\mathbf{p}$ and $\mathbf{p}$ is just a point, a “degenerate” line.)
• Algebraically, for a set $S$ to be affine, the definition requires that every affine combination of two points of $S$ belong to $S$. Remarkably, this is equivalent to requiring that $S$ contain every affine combination of an arbitrary number of points of $S$.

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PROOF

• Suppose that $S$ is affine and use induction (数学归纳法) on the number $m$ of points of $S$ occurring in an affine combination.
• (i). When $m$ is 1 or 2, an affine combination of $m$ points of $S$ lies in $S$, by the definition of an affine set.
• (ii). Now, assume that every affine combination of $k$ or fewer points of $S$ yields a point in $S$, and consider a combination of $k+1$ points. Take ${\mathbf{v}}_i$ in $S$ for $i=1,...,k+1$, and let $\mathbf{y}=c_1{\mathbf{v}}_1+...+c_k{\mathbf{v}}_k+c_{k+1}{\mathbf{v}}_{k+1}$, where $c_1+...+c_{k+1}=1$. Since the $c_i$ ’s sum to 1, at least one of them must not be equal to 1. By reindexing the ${\mathbf{v}}_i$ and $c_i$ , if necessary, we may assume that $c_{k+1}\ne 1$. Let $t=c_1+...+c_k$. Then $t=1-c_{k+1}\ne 0$, and
  
  By the induction hypothesis, the point $\mathbf{z}=(c_1/t){\mathbf{v}}_1+...+(c_k/t){\mathbf{v}}_k$ is in $S$. Thus (6) displays $\mathbf{y}$ as an affine combination of two points in $S$. By the principle of induction, every affine combination of such points lies in $S$. That is, $\text{aff }S\subset S$. The reverse inclusion, $S\subset \text{aff }S$, always applies.

Connection with subspaces of ${\mathbb{R}}^n$

Flats (平面)

translate: 平移

• For example, in ${\mathbb{R}}^3$, the proper subspaces consist of the origin $0$, the set of all lines through $0$, and the set of all planes through $0$. Thus the proper flats in ${\mathbb{R}}^3$ are points (zero-dimensional), lines (one-dimensional), and planes (two-dimensional), which may or may not pass through the origin.
• The next theorem shows that these geometric descriptions of lines and planes in ${\mathbb{R}}^3$ (as translates of subspaces) actually coincide with their earlier algebraic descriptions as sets of all affine combinations of two or three points, respectively.

Affine set $\iff$ Flat

PROOF

• Suppose that $S$ is affine. Let $\mathbf{p}$ be any fixed point in $S$ and let $W=S+(-\mathbf{p})$, so that $S=W+\mathbf{p}$. To show that $S$ is a flat, it suffices to show that $W$ is a subspace of ${\mathbb{R}}^n$.
  • Since $\mathbf{p}$ is in $S$, the zero vector is in $W$. To show that $W$ is closed under sums and scalar multiples, it suffices to show that if ${\mathbf{u}}_1$ and ${\mathbf{u}}_2$ are elements of $W$, then ${\mathbf{u}}_1+t{\mathbf{u}}_2$ is in $W$ for every real $t$. Since ${\mathbf{u}}_1$ and ${\mathbf{u}}_2$ are in $W$ , there exist ${\mathbf{s}}_1$ and ${\mathbf{s}}_2$ in $S$ such that
    Let $\mathbf{y}=s_1+s_2-\mathbf{p}$. Then $\mathbf{y}$ is an affine combination of points in $S$. Since $S$ is affine, $\mathbf{y}$ is in $S$. But then $(1-t){\mathbf{s}}_1+t\mathbf{y}$ is also in $S$. So ${\mathbf{u}}_1+t{\mathbf{u}}_2$ is in $-\mathbf{p}+S=W$. This shows that $W$ is a subspace of ${\mathbb{R}}^n$. Thus $S$ is a flat, because $S=W+\mathbf{p}$.
• Conversely, suppose $S$ is a flat. That is, $S=W+\mathbf{p}$ for some $\mathbf{p}\in {\mathbb{R}}^n$ and some subspace $W$ . To show that $S$ is affine, it suffices to show that for any pair ${\mathbf{s}}_1$ and ${\mathbf{s}}_2$ of points in $S$, the line through ${\mathbf{s}}_1$ and ${\mathbf{s}}_2$ lies in $S$. By definition of $W$, there exist ${\mathbf{u}}_1$ and ${\mathbf{u}}_2$ in $W$ such that ${\mathbf{s}}_1={\mathbf{u}}_1+\mathbf{p}$ and ${\mathbf{s}}_2={\mathbf{u}}_2+\mathbf{p}$. So, for each real $t$,
  Since $W$ is a subspace, $(1-t){\mathbf{u}}_1+t{\mathbf{u}}_2\in W$ and so $(1-t)s_1+t{s}_2\in W+\mathbf{p}=S$. Thus $S$ is affine.

Remark: Notice the key role that definitions play in this proof. For example, the first part assumes that $S$ is affine and seeks to show that $S$ is a flat. By definition, a flat is a translate of a subspace. By choosing $\mathbf{p}$ in $S$ and defining $W=S+(-\mathbf{p})$, the set $S$ is translated to the origin and $S=W+\mathbf{p}$. It remains to show that $W$ is a subspace, for then $S$ will be a translate of a subspace and hence a flat.

A geometric view of affine combinations

• Theorem 3 provides a geometric way to view the affine hull of a set: it is the flat that consists of all the affine combinations of points in the set.
  • For instance, the affine combination of two linearly independent vectors (points) is just the line through these two points; the affine combination of three linearly independent vectors (points) is the plane through these three points. (In Figure 3, ${\mathbf{b}}_1,{\mathbf{b}}_2$, and ${\mathbf{b}}_3$ is a basis of ${\mathbb{R}}^3$.
    

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EXERCISE 13

• Suppose $\{{\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3\}$ is a basis for ${\mathbb{R}}^3$. Show that $Span\{{\mathbf{v}}_2-{\mathbf{v}}_1,{\mathbf{v}}_3-{\mathbf{v}}_1\}$ is a plane in ${\mathbb{R}}^3$.

SOLUTION

• [Hint: $Span\{{\mathbf{v}}_2-{\mathbf{v}}_1,{\mathbf{v}}_3-{\mathbf{v}}_1\}$ is a plane if and only if $\{{\mathbf{v}}_2-{\mathbf{v}}_1,{\mathbf{v}}_3-{\mathbf{v}}_1\}$ is linearly independent.]

EXERCISE 14

• Show that if $\{{\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3\}$ is a basis for ${\mathbb{R}}^3$, then $\text{aff }\{{\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3\}$ is the plane through ${\mathbf{v}}_1,{\mathbf{v}}_2,$ and ${\mathbf{v}}_3$.

SOLUTION

• It suffices to show that $\text{aff }\{{\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3\}$ is the translation of a subspace. if $\mathbf{y}\in \text{aff }\{{\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3\}$, then $\mathbf{y}-{\mathbf{v}}_1\in Span\{{\mathbf{v}}_2-{\mathbf{v}}_1,{\mathbf{v}}_3-{\mathbf{v}}_1\}$. Thus $\mathbf{y}\in {\mathbf{v}}_1+Span\{{\mathbf{v}}_2-{\mathbf{v}}_1,{\mathbf{v}}_3-{\mathbf{v}}_1\}$.

EXERCISE 16

• Let $\mathbf{v}\in {\mathbb{R}}^n$ and let $k\in \mathbb{R}$. Prove that $S=\{\mathbf{x}\in {\mathbb{R}}^n:\mathbf{x}\cdot \mathbf{v}=k\}$ is an affine subset of ${\mathbb{R}}^n$.

SOLUTION

• $W=\{\mathbf{x}\in {\mathbb{R}}^n:\mathbf{x}\cdot \mathbf{v}=0\}$ is a subspace of ${\mathbb{R}}^n$. Suppose $\mathbf{p}\cdot \mathbf{v}=k$, then $S=W+\mathbf{p}$.

Connection with computer graphics

• The next theorem provides another view of affine combinations, which for ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$ is closely connected to applications in computer graphics. (See the next section and Section 2.7)

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EXAMPLE 4

• Let
  Write $\mathbf{p}$ as an affine combination of ${\mathbf{v}}_1,{\mathbf{v}}_2$, and ${\mathbf{v}}_3$, if possible.

SOLUTION

• Row reduce the augmented matrix for the equation
  To simplify the arithmetic, move the fourth row of 1’s to the top (equivalent to three row interchanges). After this, the number of arithmetic operations here is basically the same as the number needed for the method using Theorem 1.
  Thus
  

References

• $Linear$ $algebra$ $and$ $its$ $applications$