本文为《Linear algebra and its applications》的读书笔记

Hyperplanes

Hyperplanes play a special role in the geometry of R n \R^n Rn because they divide the space into two disjoint pieces, just as a plane separates R 3 \R^3 R3 into two parts and a line cuts through R 2 \R^2 R2. The key to working with hyperplanes is to use simple i m p l i c i t implicit implicit descriptions, rather than the e x p l i c i t explicit explicit or parametric representations of lines and planes used in the earlier work with affine sets.

An implicit equation of a line in R 2 \R^2 R2 has the form a x + b y = d ax + by= d ax+by=d. An implicit equation of a plane in R 3 \R^3 R3 has the form a x + b y + c z = d ax+ by+ cz= d ax+by+cz=d. Both equations describe the line or plane as the set of all points at which a linear expression (also called a l i n e a r linear linear f u n c t i o n a l functional functional (线性函数)) has a fixed value, d d d.

If f f f is a linear functional on R n \R^n Rn, then the standard matrix of this linear transformation f f f is a 1 × n 1\times n 1×n matrix A A A, say A = [ a 1 a 2 . . . a n ] A =\begin{bmatrix} a_1& a_2&...& a_n\end{bmatrix} A=[a1​​a2​​...​an​​]. So

If f f f is a nonzero functional, then r a n k A = 1 rankA = 1 rankA=1, and d i m N u l A = n − 1 dim NulA = n - 1 dimNulA=n−1. Thus, the subspace [ f : 0 ] [f: 0] [f:0] has dimension n − 1 n-1 n−1 and so is a hyperplane. Also, if d d d is any number in R R R, then

Recall that the set of solutions of A x = b A\boldsymbol x= \boldsymbol b Ax=b is obtained by translating the solution set of A x = 0 A\boldsymbol x=\boldsymbol 0 Ax=0. Then

Thus the sets [ f : d ] [f: d] [f:d] are hyperplanes parallel to [ f : 0 ] [f: 0] [f:0]. See Figure 1.

When A A A is a 1 × n 1\times n 1×n matrix, the equation A x = d A\boldsymbol x = d Ax=d may be written with an inner product n ⋅ x \boldsymbol n\cdot\boldsymbol x n⋅x, using n \boldsymbol n n in R n \R^n Rn with the same entries as A A A. Thus, from (2),

Then [ f : 0 ] = { x ∈ R n : n ⋅ x = 0 } [f: 0]=\{\boldsymbol x\in R^n:\boldsymbol n\cdot \boldsymbol x=0\} [f:0]={x∈Rn:n⋅x=0}, which shows that [ f , 0 ] [f,0] [f,0] is the orthogonal complement of the subspace spanned by n \boldsymbol n n. In the terminology of calculus and geometry for R 3 \R^3 R3, n \boldsymbol n n is called a normal vector (法向量) to [ f : 0 ] [f:0] [f:0]. (A “normal” vector in this sense need not have unit length.) Also, n \boldsymbol n n is said to be normal to each parallel hyperplane [ f : d ] [f:d] [f:d], even though n ⋅ x \boldsymbol n\cdot x n⋅x is not zero when d ≠ 0 d\neq 0 d​=0.

Another name for [ f : d ] [f:d] [f:d] is a level set (水平集) of f f f , and n n n is sometimes called the gradient (梯度) of f f f when f ( x ) = n ⋅ x f(\boldsymbol x)= \boldsymbol n\cdot \boldsymbol x f(x)=n⋅x for each x \boldsymbol x x.

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The next three examples show connections between implicit and explicit descriptions of hyperplanes.

EXAMPLE 4
In R 2 \R^2 R2, give an explicit description of the line x − 4 y = 13 x-4y=13 x−4y=13 in parametric vector form.
SOLUTION

EXAMPLE 5
Let

, and let L 1 L_1 L1​ be the line through v 1 \boldsymbol v_1 v1​ and v 2 \boldsymbol v_2 v2​. Find a linear functional f f f and a constant d d d such that L 1 = [ f : d ] L_1=[f:d] L1​=[f:d].
SOLUTION
The line L 1 L_1 L1​ is parallel to the translated line L 0 L_0 L0​ through v 2 − v 1 \boldsymbol v_2-\boldsymbol v_1 v2​−v1​ and the origin. The defining equation for L 0 L_0 L0​ has the form

Since n \boldsymbol n n is orthogonal to the subspace L 0 L_0 L0​, which contains v 2 − v 1 \boldsymbol v_2-\boldsymbol v_1 v2​−v1​, then

By inspection, a solution is [    a     b    ] = [    2     5    ] [\ \ a\ \ \ b\ \ ]= [\ \ 2\ \ \ 5\ \ ] [  a   b  ]=[  2   5  ]. Let f ( x , y ) = 2 x + 5 y f(x, y) =2x+5y f(x,y)=2x+5y. From (5), L 0 = [ f : 0 ] L_0=[f:0] L0​=[f:0], and L 1 = [ f : d ] L_1=[f:d] L1​=[f:d] for some d d d. Since v 1 \boldsymbol v_1 v1​ is on line L 1 L_1 L1​, d = f ( v 1 ) = 2 ( 1 ) + 5 ( 2 ) = 12 d=f(\boldsymbol v_1)=2(1) +5(2)=12 d=f(v1​)=2(1)+5(2)=12. Thus, the equation for L 1 L_1 L1​ is 2 x + 5 y = 12 2x + 5y = 12 2x+5y=12.

EXAMPLE 6
Let

Find an implicit description [ f : d ] [f:d] [f:d] of the plane H 1 H_1 H1​ that passes through v 1 , v 2 , \boldsymbol v_1, \boldsymbol v_2, v1​,v2​, and v 3 \boldsymbol v_3 v3​.
SOLUTION
H 1 H_1 H1​ is parallel to a plane H 0 H_0 H0​ through the origin that contains the translated points

Since these two points are linearly independent, H 0 = S p a n { v 2 − v 1 , v 3 − v 1 } H_0= Span\{\boldsymbol v_2-\boldsymbol v_1, \boldsymbol v_3-\boldsymbol v_1\} H0​=Span{v2​−v1​,v3​−v1​}. Let n = [ a b c ] \boldsymbol n=\begin{bmatrix} a\\b\\c\end{bmatrix} n=⎣⎡​abc​⎦⎤​ be the normal to H 0 H_0 H0​. Then v 2 − v 1 \boldsymbol v_2- \boldsymbol v_1 v2​−v1​ and v 3 − v 1 \boldsymbol v_3 -\boldsymbol v_1 v3​−v1​ are each orthogonal to n \boldsymbol n n:

These two equations form a system whose augmented matrix can be row reduced:

Row operations yield

Set c = 4 c= 4 c=4, for instance. Then n = [ − 2 5 4 ] \boldsymbol n=\begin{bmatrix} -2\\5\\4\end{bmatrix} n=⎣⎡​−254​⎦⎤​ and H 0 = [ f : 0 ] H_0=[f:0] H0​=[f:0], where f ( x ) = − 2 x 1 + 5 x 2 + 4 x 3 f(\boldsymbol x)=-2x_1+5x_2 + 4x_3 f(x)=−2x1​+5x2​+4x3​.

The parallel hyperplane H 1 H_1 H1​ is [ f : d ] [f :d] [f:d]. To find d d d, use the fact that v 1 \boldsymbol v_1 v1​ is in H 1 H_1 H1​, and compute d = f ( v 1 ) = f ( 1 , 1 , 1 ) = 7 d = f(\boldsymbol v_1)= f(1, 1, 1)= 7 d=f(v1​)=f(1,1,1)=7.

The procedure in Example 6 generalizes to higher dimensions. However, for the
special case of R 3 \R^3 R3, one can also use the cross-product formula (叉积公式) to compute n n n, using a symbolic determinant as a mnemonic device:

If only the formula for f f f is needed, the cross-product calculation may be written as an ordinary determinant:

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PROOF
Suppose that H H H is a hyperplane, take p ∈ H \boldsymbol p\in H p∈H, and let H 0 = H − p H_0= H -\boldsymbol p H0​=H−p. Then H 0 H_0 H0​ is an ( n − 1 ) (n-1) (n−1)-dimensional subspace. Next, take any point y \boldsymbol y y that is not in H 0 H_0 H0​. By the Orthogonal Decomposition Theorem,

where y 1 \boldsymbol y_1 y1​ is a vector in H 0 H_0 H0​ and n \boldsymbol n n is orthogonal to every vector in H 0 H_0 H0​. The function f f f defined by

is a linear functional, by properties of the inner product. Now, [ f : 0 ] [f :0] [f:0] is a hyperplane that contains H 0 H_0 H0​, by construction of n \boldsymbol n n. It follows that

Finally, let d = f ( p ) = n ⋅ p d= f(\boldsymbol p)=\boldsymbol n\cdot \boldsymbol p d=f(p)=n⋅p. Then, as in (3) shown earlier,

The converse statement that [ f : d ] [f :d] [f:d] is a hyperplane follows from (1) and (3) above.

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Many important applications of hyperplanes depend on the possibility of “separating” two sets by a hyperplane. The following terminology and notation will help to make this idea more precise.

Topology: 拓扑
open ball: 开球
A set is open: 开集
A set is closed: 闭集
A set is bounded: 有界集
A set is compact: 紧致集

EXERCISE 27
Give an example of a closed subset S S S of R 2 \R^2 R2 such that c o n v S conv S convS is not closed.
SOLUTION
S = { p ∣ p = ( x , y ) , y = 1 / x , x ≥ 1 / 2 } S=\{\boldsymbol p|\boldsymbol p=(x,y),y=1/x,x\geq1/2\} S={p∣p=(x,y),y=1/x,x≥1/2}

EXERCISE 29
Prove that the open ball B ( p , δ ) = { x : ∥ x − p ∥ < δ } B(\boldsymbol p,\delta)=\{\boldsymbol x:\left\|\boldsymbol x-\boldsymbol p\right\|<\delta\} B(p,δ)={x:∥x−p∥<δ} is a convex set.
SOLUTION
[Hint: Use the Triangle Inequality.] (三角不等式)

EXAMPLE 7
Let

as shown in Figure 3. Then the set S S S is closed since it contains all its boundary points. The set S S S is bounded since S ⊂ B ( 0 , 3 ) S\subset B(\boldsymbol 0, 3) S⊂B(0,3). Thus S S S is also compact.

N o t a t i o n Notation Notation: If f f f is a linear functional, then f ( A ) ≤ d f(A)\leq d f(A)≤d means f ( x ) ≤ d f(\boldsymbol x)\leq d f(x)≤d for each x ∈ A \boldsymbol x\in A x∈A.

strictly seperate: 严格分割

Notice that strict separation requires that the two sets be disjoint, while mere separation does not.

PROOF
Suppose that ( c o n v A ) ∩ ( c o n v B ) = ϕ (conv A)\cap(convB)=\phi (convA)∩(convB)=ϕ. Since the convex hull of a compact set is compact, Theorem 12 ensures that there is a hyperplane H H H that strictly separates c o n v A convA convA and c o n v B convB convB. Clearly, H H H also strictly separates the smaller sets A A A and B B B.

Conversely, suppose the hyperplane H = [ f : d ] H =[f :d] H=[f:d] strictly separates A A A and B B B. Without loss of generality, assume that f ( A ) < d f(A) < d f(A)<d and f ( B ) > d f(B) > d f(B)>d. Let x = c 1 x 1 + . . . + c k x k \boldsymbol x = c_1\boldsymbol x_1+...+ c_k\boldsymbol x_k x=c1​x1​+...+ck​xk​ be any convex combination of elements of A A A. Then

Thus f ( c o n v A ) < d f(conv A) < d f(convA)<d. Likewise, f ( c o n v B ) > d f(convB) > d f(convB)>d, so H = [ f : d ] H=[f :d] H=[f:d] strictly separates c o n v A convA convA and c o n v B convB convB. By Theorem 12, c o n v A convA convA and c o n v B convB convB must be disjoint.

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EXERCISE 14
Let F 1 F_1 F1​ and F 2 F_2 F2​ be 4-dimensional flats in R 6 \R^6 R6, and suppose that F 1 ∩ F 2 ≠ ϕ F_1\cap F_2 \neq\phi F1​∩F2​​=ϕ. What are the possible dimensions of F 1 ∩ F 2 F_1\cap F_2 F1​∩F2​?
SOLUTION

下面的答案是我自己写的，感觉论证啰嗦且不太严谨，仅供参考
如果有好的解答，欢迎一起交流~

Let F 1 = W 1 + p 1 , F 2 = W 2 + p 2 F_1=W_1+\boldsymbol p_1,F_2=W_2+\boldsymbol p_2 F1​=W1​+p1​,F2​=W2​+p2​, where W 1 , W 2 W_1,W_2 W1​,W2​ are two 4-dimensional subspaces. Suppose a 1 , . . . , a 4 \boldsymbol a_1,...,\boldsymbol a_4 a1​,...,a4​ and b 1 , . . . , b 4 \boldsymbol b_1,...,\boldsymbol b_4 b1​,...,b4​ be the basis of W 1 W_1 W1​ and W 2 W_2 W2​ respectively. Let F 1 ∩ F 2 = W + p F_1\cap F_2=W+\boldsymbol p F1​∩F2​=W+p and x ∈ W \boldsymbol x\in W x∈W, then there exist m i , n i ∈ R m_i,n_i\in R mi​,ni​∈R ( 1 ≤ i ≤ 4 1\leq i\leq4 1≤i≤4) such that
p 1 + m 1 a 1 + . . . + m 4 a 4 = p 2 + n 1 b 1 + . . . + n 4 b 4 = x + p p 1 − p + m 1 a 1 + . . . + m 4 a 4 = p 2 − p + n 1 b 1 + . . . + n 4 b 4 = x     ( 1 ) \boldsymbol p_1+m_1\boldsymbol a_1+...+m_4\boldsymbol a_4=\boldsymbol p_2+n_1\boldsymbol b_1+...+n_4\boldsymbol b_4=\boldsymbol x+\boldsymbol p\\ \boldsymbol p_1-\boldsymbol p+m_1\boldsymbol a_1+...+m_4\boldsymbol a_4=\boldsymbol p_2-\boldsymbol p+n_1\boldsymbol b_1+...+n_4\boldsymbol b_4=\boldsymbol x\ \ \ (1) p1​+m1​a1​+...+m4​a4​=p2​+n1​b1​+...+n4​b4​=x+pp1​−p+m1​a1​+...+m4​a4​=p2​−p+n1​b1​+...+n4​b4​=x   (1)

Notice that d i m F 1 ∩ F 2 = d i m W dimF_1\cap F_2=dimW dimF1​∩F2​=dimW ( d i m W dimW dimW is the dimension of the solution set of x \boldsymbol x x).

Since x \boldsymbol x x can be 0 \boldsymbol 0 0, there exist m i ′ , n i ′ ∈ R m_i',n_i'\in R mi′​,ni′​∈R ( 1 ≤ i ≤ 4 1\leq i\leq4 1≤i≤4) such that

p 1 + m 1 ′ a 1 + . . . + m 4 ′ a 4 = p 2 + n 1 ′ b 1 + . . . + n 4 ′ b 4 = p     ( 2 ) \boldsymbol p_1+m_1'\boldsymbol a_1+...+m_4'\boldsymbol a_4=\boldsymbol p_2+n_1'\boldsymbol b_1+...+n_4'\boldsymbol b_4=\boldsymbol p\ \ \ (2) p1​+m1′​a1​+...+m4′​a4​=p2​+n1′​b1​+...+n4′​b4​=p   (2)

From ( 2 ) (2) (2), we know that p 1 − p \boldsymbol p_1-\boldsymbol p p1​−p is a linear combination of { a 1 , . . . , a 4 } \{\boldsymbol a_1,...,\boldsymbol a_4\} {a1​,...,a4​} and p 2 − p \boldsymbol p_2-\boldsymbol p p2​−p is a linear combination of { b 1 , . . . , b 4 } \{\boldsymbol b_1,...,\boldsymbol b_4\} {b1​,...,b4​}.Thus by ( 1 ) (1) (1), there exist t i , s i ∈ R t_i,s_i\in R ti​,si​∈R ( 1 ≤ i ≤ 4 1\leq i\leq4 1≤i≤4) such that
t 1 a 1 + . . . + t 4 a 4 = s 1 b 1 + . . . + s 4 b 4 = x     ( 3 ) t 1 a 1 + . . . + t 4 a 4 − s 1 b 1 − . . . − s 4 b 4 = 0     ( 4 ) t_1\boldsymbol a_1+...+t_4\boldsymbol a_4=s_1\boldsymbol b_1+...+s_4\boldsymbol b_4=\boldsymbol x\ \ \ (3)\\ t_1\boldsymbol a_1+...+t_4\boldsymbol a_4-s_1\boldsymbol b_1-...-s_4\boldsymbol b_4=\boldsymbol 0\ \ \ (4) t1​a1​+...+t4​a4​=s1​b1​+...+s4​b4​=x   (3)t1​a1​+...+t4​a4​−s1​b1​−...−s4​b4​=0   (4)

Let A = [ a 1 . . . a 4 b 1 . . . b 4 ] A=\begin{bmatrix}\boldsymbol a_1&...&\boldsymbol a_4&\boldsymbol b_1&...&\boldsymbol b_4\end{bmatrix} A=[a1​​...​a4​​b1​​...​b4​​], then according to ( 4 ) (4) (4):
A [ t 1 . . . t 4 − s 1 . . . − s 4 ] = 0 ∵ 4 ≤ r a n k A ≤ 6 , d i m N u l A = 8 − r a n k A ∴ 2 ≤ d i m N u l A ≤ 4 A\begin{bmatrix}t_1\\...\\t_4\\-s_1\\...\\-s_4\end{bmatrix}=\boldsymbol 0\\\because 4\leq rankA\leq6,dimNulA=8-rankA\\\therefore 2\leq dimNulA\leq 4 A⎣⎢⎢⎢⎢⎢⎢⎡​t1​...t4​−s1​...−s4​​⎦⎥⎥⎥⎥⎥⎥⎤​=0∵4≤rankA≤6,dimNulA=8−rankA∴2≤dimNulA≤4

According to ( 3 ) (3) (3), it can be shown that the dimension of the solution set of x \boldsymbol x x and [ t 1 . . . t 4 ] \begin{bmatrix}t_1\\...\\t_4\end{bmatrix} ⎣⎡​t1​...t4​​⎦⎤​ are equal. We can also observe that [ t 1 . . . t 4 ] ↦ [ s 1 . . . s 4 ] \begin{bmatrix}t_1\\...\\t_4\end{bmatrix}\mapsto\begin{bmatrix}s_1\\...\\s_4\end{bmatrix} ⎣⎡​t1​...t4​​⎦⎤​↦⎣⎡​s1​...s4​​⎦⎤​ is a linear transformation, thus d i m N u l A = dimNulA= dimNulA= the dimension of the solution set of [ t 1 . . . t 4 ] \begin{bmatrix}t_1\\...\\t_4\end{bmatrix} ⎣⎡​t1​...t4​​⎦⎤​. So d i m F 1 ∩ F 2 = d i m N u l A dimF_1\cap F_2=dimNulA dimF1​∩F2​=dimNulA and 2 ≤ d i m F 1 ∩ F 2 ≤ 4 2\leq dimF_1\cap F_2\leq4 2≤dimF1​∩F2​≤4.