Linear Algebra Notes

Linear Algebra Done Right

• 1.18 定义加法和乘法

  • An addition on a set V is a function that assigns an element u+v ∈ V to each pair of elements u, v ∈ V.
  • A scalar multiplication on a set V is a function that assigns an element λv ∈ V to each λ ∈ F and each v ∈ V.

• 1.19 向量空间 (又叫线性空间)

  域\(F\)上的向量空间是一个集合\(V\)，\(V\)上有两个运算\(+ : V × V → V\)(加法)和\(· : F × V → V\)(标量乘法)，满足以下性质：

  公理	说明
  向量加法的结合律	u + (v + w) = (u + v) + w
  向量加法的交换律	u + v = v + u
  向量加法的单位元	存在一个叫做零向量的元素0 ∈ V，使得对任意u ∈ V都满足u + 0 = u
  向量加法的逆元素	对任意v ∈ V都存在其逆元素−v ∈ V使得v + (−v) = 0
  标量乘法与标量的域乘法相容	a(bv) = (ab)v, a,b ∈ F
  标量乘法的单位元	域F存在乘法单位元1满足1v = v
  标量乘法对向量加法的分配律	a(u + v) = au + av
  标量乘法对域加法的分配律	(a + b)v = av + bv

  注意这里加法和乘法自带要求运算是封闭的。

• 1.20 定义 vector, point

Elements of a vector space are called vectors or points.

• 1.21 Definition real vector space, complex vector space

  • A vector space over R is called a real vector space.
  • A vector space over C is called a complex vector space.
• 1.22 例：\(\mathbb{F}^\infty\) 是 \(\mathbb{F}\) 上的向量空间
• 1.23 例：\(f,g\)是集合S到\(F\)是的函数，f+g := \x → f(x)+g(x)，λf := \x → λ(f(x))，则全体这样的函数是向量空间
• 1.25 性质：向量的加法零元是唯一的
• 1.26 性质：向量的加法逆元是唯一的
• 1.29 性质：标量0乘以向量等于0向量
• 1.30 性质：向量0乘以标量等于0向量

到此基本上是说向量的加减乘除是intuitive的

• 1.32 定义 子空间 (subspace)

若U是V的子集，且U用V的加法和乘法也是一个向量空间，则称U是V的子空间。

• 1.33 例子 \(\left\{ (x_1, x_2, 0) : x_1, x_2 \in \mathbb{F} \right\}\) 是 \(\mathbb{F}^3\) 的子空间
• 1.34 子空间成立的条件

U是V的子空间(subspace)当且仅当以下三个条件成立

• additive identitty \(\boldsymbol{0} \in \boldsymbol{U}\)
• closed under addition \(x,y \in \boldsymbol{U} \implies x+y \in \boldsymbol{U}\)
• closed under scalar multiplication \(a \in \mathbb{F} \text{ and } \boldsymbol{u} \in \boldsymbol{U} \implies a \boldsymbol{u} \in \boldsymbol{U}\)

证：正向trivial，反向：由定义也不难证，略了

• 1.36 定义 sum of subspaces

  Suppose \(\boldsymbol{U_1} \cdots \boldsymbol{U}_m\) are subsets of \(\boldsymbol{V}\).

  \[\boldsymbol{U}_1 + \cdots + \boldsymbol{U}_m = \left\{ \boldsymbol{u}_1 + \cdots + \boldsymbol{u}_m : \boldsymbol{u}_1 \in \boldsymbol{U}_1, \cdots, \boldsymbol{u}_m \in \boldsymbol{U}_m \right\}\]

• 1.39 定理 Sum of subspaces 是包含所有subspaces中最小的

  证：显然 Sum of subspaces 是 subspace!，又显然 Sum of subspaces 包含了 \(\boldsymbol{U}_1,\, \boldsymbol{U}_2,\, \cdots\)，所以 Sum of subspaces足够大

  又由space的封闭性，任何包含\(\boldsymbol{U}_1,\, \boldsymbol{U}_2,\, \cdots\)的又都包含 Sum of subspaces，所以 Sum of subspaces足够小

1.40 定义 direct sum

Suppose \(\boldsymbol{U_1} \cdots \boldsymbol{U}_m\) are subsets of \(\boldsymbol{V}\).

• The sum is called a direct sum if each element of \(\boldsymbol{U_1} \cdots \boldsymbol{U}_m\) can be written in only one way as a sum \(\boldsymbol{u}_1 + \cdots + \boldsymbol{u}_m\) , where each \(\boldsymbol{u}_j \in \boldsymbol{U}_j\)
• Direct sum is denoted by \(\boldsymbol{U_1} \oplus \cdots \oplus \boldsymbol{U}_m\)

有点类似于正交子空间，也是笛卡尔积。

• 1.44 direct sum 等价定义为 0 只有一种表示法

• 1.45 两个子空间的 direct sum 等价定义为它们的交集只包含一个0元素

• 2.3 定义 linear combination 一组向量的线性组合
• 2.5 定义 名词span
• 2.7 定理 一组向量的span是包含这组向量的最小的subspace
• 2.8 定义 动词span
• 2.10 定义 有限维向量空间：有限个向量张成的空间为有限维向量空间

• 2.11 定义 polynomial: A function p : F → F is called a polynomial with coefficients in F if there exist \(a_0, \cdots, a_m \in \mathbb{F}\) such that

  \[p(z) = a_0 + a_1z + a_2z^2 + \cdots + a_mz^m\]

  for all \(z \in \boldsymbol{F}\).

  \(\mathcal{P}(\mathbb{F})\) is the set of all polynomials with coefficients in \(\mathbb{F}\).

• 2.12 定义 Degree of polynomial 是最高项次数，若为0则定义为\(-\infty\)
• 2.13 定义 \(\mathcal{P}_m(\mathbb{F})\)是最高次不超过m的多项式(m为非负整数)
• 2.14 例子 \(\mathcal{P}_m(\mathbb{F})\) 是 \(\mathbb{F}\) 上的有限维向量空间
• 2.16 例子 \(\mathcal{P}(\mathbb{F})\) 是 \(\mathbb{F}\) 上的无限维向量空间
• 2.17 定义 一组向量的线性无关：若它们的线性和为零就意味着系数都为零(若一组向量的个数为0，也认为是线性无关)
• 2.19 定义 线性相关
• 2.23 线性无关的一组向量的个数小于能张成整个空间的向量的个数
• 2.26 有限维空间的子空间是有限维空间
• 2.27 定义 Basis 基底：线性无关且张成整个空间
• 2.29 基意味着空间中的每个向量都是唯一的基底的线性组合
• 2.31 每个可张成整个空间的一组向量都包含一组基

• 2.32 每个有限维空间都存在一组基

  注意：维基说：在选择公理成立的条件下，每个无限维空间也都存在一组基，但是作者这里避开了无限维的讨论。

• 2.33 (有限维时)每个线性无关的一组向量都可以扩展成一组基(可能通过增加新的向量)

• 2.34 (有限维时)每个子空间都可以通过 direct sum (类似笛卡尔积) 得到原空间

  Suppose \(V\) is finite-dimensional and \(U\) is a subspace of V. Then there is a subspace \(W\) of V such that \(V = U \oplus W\).

• 2.35 每个有限维空间的基包含的向量的个数必相等
• 2.36 有限空间的维度：任意一组基的个数
• 2.38 有限空间的子空间的维度小于等于该有限空间的维度
• 2.39 任意适当个数的线性无关的一组向量即是有限维空间的一组基

• 2.43 Dimension of a sum 若 \(\boldsymbol{U}_1,\, \boldsymbol{U}_2\)是一有限维空间的子空间，则

  \[\text{dim}(\boldsymbol{U}_1 + \boldsymbol{U}_2) = \text{dim}\boldsymbol{U}_1 + \text{dim}\boldsymbol{U}_2 - \text{dim} \left( \boldsymbol{U}_1 \cap \boldsymbol{U}_2 \right)\]

第三章

注意这里在讨论线性变换时又把对维度的限制去掉了，可以是无穷维的。

• 3.1 Notation

  • \(\mathbb{F}\) 是 \(\mathbb{R}\) 或 \(\mathbb{C}\)
  • \(\boldsymbol{V}\) 和 \(\boldsymbol{W}\) 是 \(\mathbb{F}\) 上的向量空间

• 3.2 定义 Linear Map

  A Linear Map is a function T : V → W that satisfies

  1. additivity \(T(\boldsymbol{u} + \boldsymbol{v}) = T \boldsymbol{u} + T \boldsymbol{v}\).
  2. homogeneity \(T(\lambda \boldsymbol{v}) = \lambda(T \boldsymbol{v})\)

• 3.3 Notation

  所有V到W的线性映射记为 \(\mathcal{L}(V,W)\)

• 3.5 Linear maps and basis of domain

  若 \(L \in \mathcal{L}(V,W)\)， \(\boldsymbol{v}_i\) 是 \(V\) 的一组基， \(\boldsymbol{w}_i\) 是 \(W\) 中的一组向量(与 \(\boldsymbol{v}_i\) 个数相同)，则

  存在唯一的线性映射，将 \(\boldsymbol{v}_i\) 映射成 \(\boldsymbol{w}_i\)

  证明见书

• 3.6 定义 线性映射函数之间的加法和数乘

  若 \(S,T \in \mathcal{L}(V,W),\text{ and } \lambda \in \mathbb{F}\) 则

  \(S + T := \backslash \boldsymbol{v} → S \boldsymbol{v} + T \boldsymbol{v}\)
  \(\lambda S = \backslash \boldsymbol{v} → \lambda S \boldsymbol{v}\)

• 3.7 \(\mathcal{L}(V,W)\) 是一个向量空间。

  由于这种加法和乘法满足向量空间的公理

• 3.8 Product of Linear Maps

  \(T \in \mathcal{L}(U,V)\) and \(S \in \mathcal{L}(V,W)\)，则定义

  \(ST = \backslash \boldsymbol{u} → S ( T ( u ) )\) for \(\boldsymbol{u} \in U\)

• 3.9 Algebraic properties of products of linear maps

  • associativity \((T_1T_2)T_3 = T_1(T_2T_3)\)
  • identity \(TI_V = I_WT = T\) 注： \(T \in \mathcal{L}(V,W)\).
  • distributive properties \((S_1 + S_2)T = S_1 T + S_2 T,\, S(T_1+T_2) = ST_1 + ST_2\)

  书上把证明留给读者了，我也略了。

• 习题 3A.8

  Give an example of a function φ: ℝ² → ℝ such that φ(av) = aφ(v) 但 φ 不是线性的

  φ 是无穷范数, φ(x) = | max(x₁, x₂) |, φ([0,1]) + φ([1,0]) = 2, φ([1,1]) = 1

• 习题 3A.9

  Give an example of a function φ: ℂ² → ℂ such that φ(w+z)=φ(w)+φ(z) 但 φ 不是线性的

  (Here C is thought of as a complex vector space.) [There also exists such function in ℝ. However, showing the existence of such a function involves considerably more advanced tools.]

  φ(x) = Re(x), φ(i⋅1) = 0, i⋅φ(1) = i

• 3.12 定义 Null space, null T

  For \(T \in \mathcal{L}(V,W)\), the null space of T, denoted null T, is the subset of V consisting of those vectors that T maps to 0:

  \[\text{ null } T = \left\{ v \in V : T(v)=0 \right\}\]

• 3.14 定义 Null space is a subspace

  首先，0在null T里，由T的线性性质，可得null T里的元素也对加法和数乘封闭，所以null T满足0元，加法封闭，乘法封闭这三个判定条件。

• 3.15 定义 injective (one-to-one) 单射

  A function T: V -> W is called injective if T(u)=T(v) implies u=v.

• 3.16 定理 Injectivity is equivalent to null space equals {0}

  证明比较Trivial，略

• 3.17 定义 Range 值域
• 3.19 定理 Range is a subspace
• 3.20 定义 surjective (onto) (满射)

第三章

注意：在讨论维度和进行计算时，又把有限维的要求加上了。

• 3.22 定理 Fundamental Theorem of Linear Algebra 线性代数基本定理 (只是一部分)

  若 \(V\) 是有限维向量空间，\(T \in \mathcal{L}(V,W)\)，则T的值域是有限维的，且

  \[\text{dim }V = \text{dim null }T + \text{dim range }T\]

  注意：由2.34，V可以直接可以被分成两个子空间(正交子空间)，W也是。

• 3.23 映射到更小维度空间的映射不是单射(injective)

  dim null T = dim V - dim range T = dim V - dim W > 0;

• 3.24 映射到更大维度空间的映射不是满射(surjective)

• 3.30 定义 matrix, \(A_{j,k}\)

  Let m and n denote positive integers. An m-by-n matrix A is a rectangular array of elements of F with m rows and n columns:

  \[A = \begin{pmatrix} A_{1,1} & \cdots & A_{1,n} \\ \vdots & \ddots & \vdots \\ A_{m,1} & \cdots & A_{m,n} \end{pmatrix}\]

  The notation \(A_{j,k}\) denotes the entry in row j , column k of A.

• 3.32 定义 matrix of a linear map, \(\mathcal{M}(T)\) Suppose \(T \in \mathcal{L}(V,W)\) and \(v_1, \cdots, v_n\) is a basis of \(V\) and \(w_1, \cdots, w_m\) is a basis of \(W\). The matrix of T with respect to these bases is the m-by-n matrix \(\mathcal{M}(T)\) whose entries \(A_{j,k}\) are defined by

  \[T v_k = A_{1,k}w_1 + \cdots + A_{m,k}w_m\]

  If the bases are not clear from the context, then the notation \(\mathcal{M}\left(T, \left(v_1, \cdots, v_n\right), \left(w_1, \cdots, w_m\right)\right)\) is used

  注： \(\boldsymbol{w} = (\mathcal{M}(T)) \boldsymbol{v}\)

• 3.35 定义 矩阵相加是矩阵元素的element-wise加
• 3.36 定理 \(\mathcal{M}(S+T) = \mathcal{M}(S) + \mathcal{M}(T)\)
• 3.37 定义 矩阵的数乘
• 3.38 定理 \(\mathcal{M}(\lambda T) = \lambda \mathcal{M}(T)\)

• 3.39 Notation \(\mathbb{F}^{m,n}\)

  For m and n positive integers, the set of all m-by-n matrices with entries in \(\mathbb{F}\) is denoted by \(\mathbb{F}^{m,n}\)

• 3.40 定理 \(\mathbb{F}^{m,n}\) 是向量空间， \(\text{dim }\mathbb{F}^{m,n} = mn\)
• 3.41 定义 矩阵乘法
• 3.43 定理 \(\mathcal{M}(S \circ T) = \mathcal{M}(S) \mathcal{M}(T)\)
• 3.44 Notation \(A_{j,\cdot},\, A_{\cdot,k}\)
• 3.47 \((AC)_{j,k} = A_{j,\cdot}C_{\cdot,k}\)
• 3.49 \((AC)_{\cdot,k} = AC_{\cdot,k},\, (AC)_{j,\cdot} = A_{j,\cdot}C\)

• 3.53 定义 invertible, inverse

  A linear map \(T \in \mathcal{L}(V,W)\) is called invertible if there exists a linear map \(S \in \mathcal{L}(W,V)\) such that \(ST\) equals the identity map on \(V\) and \(TS\) equals the identity map on \(W\).

  Such \(S\) is called an inverse of \(T\).

• 3.54 定理 Inverse is unique
• 3.55 Notation 若\(T\)可逆，记它的逆为 \(T^{-1}\)
• 3.56 A linear map is invertible if and only if it’s injective and surjective. (一一映射，又叫双射，又叫bijective)

• 3.58 定义 Isomorphism, isomorphic 同构

  • An isomorphism is an invertible linear map.
  • Two vector spaces are called isomorphic if there is an isomorphism from one vector space onto the other one.

• 3.59 Dimension shows whether vector spaces are isomorphic

  Two finite-dimensional vector spaces over F are isomorphic if and only if they have the same dimension.

• 3.60 \(\mathcal{L}(V,W)\) 和 \(\mathbb{F}^{m,n}\) 同构，由矩阵 \(\mathcal{M}\) 联系起来。

  证明：略

• 3.61 \(\text{dim }\mathcal{L}(V,W) = (\text{dim }(V)) (\text{dim }(W))\)

• 3.62 定义 Matrix of a vector \(\mathcal{M}(V)\)

  Suppose \(v \in V\) and \(v_1, \cdots, v_n\) is a basis of \(V\). The matrix of v respect to this basis is the n-by-1 matrix

  \[\mathcal{M}(v) = \begin{pmatrix} c_1 \\ \vdots \\ c_n \end{pmatrix}\]

  where \(v = c_1v_1 + \cdots + c_nv_n\)

• 3.64 \(\mathcal{M}(T)_{\cdot,k} = \mathcal{M}(Tv_k)\) Note there is an printing error in the book.

  意思是一个线性变换对应的矩阵的第k列 是 原空间中第k的基的线性变换后的向量在值空间中的坐标

• 3.65 Linear maps act like matrix multiplication

  Suppose \(T \in \mathcal{L}(V,W),\, v \in V\). \(v_1, \cdots, v_n\) is a basis of \(V\) and \(w_1, \cdots, w_m\) is a basis of \(W\). Then

  \[\mathcal{M}(Tv) = \mathcal{M}(T) \mathcal{M}(v)\]

  之前 3.43 定理 \(\mathcal{M}(S \circ T) = \mathcal{M}(S) \mathcal{M}(T)\) 都讲过了，所以这3.65是啥意思？

• 3.67 定义 operator, \(\mathcal{L}(V)\)

  • A linear map from a vector space to itself is called an operator.
  • The notation \(\mathcal{L}(V)\) denotes the set of all operators on V. i.e. \(\mathcal{L}(V) = \mathcal{L}(V,V)\).

• 3.69 定理 有限维空间中单射(injective)，双射(bijective)，满射(surjective)等价。

• 3.71 定义 product of vector spaces

  Suppose \(V_1, \cdots, V_m\) are vector spaces over \(\mathbb{F}\).

  The product \(VS1 \times \cdots \times V_m\) is defined by

  \[V_1 \times \cdots \times V_M = \left\{ ( v_1 , \ldots , v_m ) : v_1 \in V_1 , \ldots , v_m \in V_M \right\}.\]

  Addition on \(V_1 \times \cdots \times V_M\) is defined by

  \[(u_1 , \ldots , u_m) + (v_1 , \ldots , v_m) = (u_1 + v_1 , \ldots , u_m + v_m ).\]

  Scalar multiplication on \(V_1 \times \cdots \times V_M\) is defined by

  \[\lambda (v_1 , \ldots , v_m ) = (\lambda v_1 , \ldots , \lambda v_m ).\]

• 3.73 Product of vector spaces is a vector space

  Suppose \(V_1, \ldots, V_M\) are vector spaces over F. Then \(V_1 \times \cdots \times V_M\) is a vector space over F.

• 3.75 Example Find a basis of \(\mathcal{P}_2(\mathbb{R}) \times \mathbb{R}^2\)

  Solution

  \[\left(1, (0, 0)\right), \left(x, (0, 0)\right), \left(x^2, (0, 0)\right), \left(0, (1, 0)\right), \left(0, (0, 1)\right).\]
• 3.76 Dimension of a product is the sum of dimensions
• 3.77 Products and direct sums are isomorphic 直和和笛卡尔积同构

• 3.78 A sum is a direct sum if and only if dimensions add up

  直和成立等价于维度不减

商空间

• 3.79 定义 v+U

  Suppose \(v \in V\) and \(U\) is a subspace of \(V\). Then \(v + U\) is the subset of \(V\) defined by

  \[v + U = \left\{v + u : u \in U \right\}.\]

• 3.81 定义 affine subset, parallel

  An affine subset of \(V\) is a subset of \(V\) of the form \(v + U\) for some \(v \in V\) and some subspace \(U\) of \(V\).

  the affine subset \(v + U\) is said to be parallel to \(U\).

• 补充一点，这里要开始说商空间了，但是

• 3.83 定义 quotient space 商空间

  Suppose \(U\) is a subspace of \(V\). Then the quotient space \(V / U\) is the set of all affine subsets of \(V\) parallel to \(U\). In other words,

  \[V / U = \left\{v + U : v \in V \right\}.\]

  接下来显示是也要把商空间搞成一个向量空间。

• 3.85 Two affine subsets parallel to U are equal or disjoint

  Suppose \(U\) is a subspace of \(V\) and \(v, w \in V\). Then the following are equivalent:

  • \(v - w \in U\).
  • \(v + U = w + U\).
  • \((v + U) \cap (w + U) \neq \emptyset\).

  这几条就说明了两个affine subset不是相等就是相离

• 3.86 定义 商空间上的加法和乘法

  Suppose \(U\) is a subspace of \(V\). Then addition and scalar multiplication are defined on \(V / U\) by

  • \((v + U) + (w + U) = (v + w) + U\).
  • \(\lambda (v + U) = (\lambda v) + U\).

  for \(v, w \in V\) and \(\lambda \in \mathbb{F}\).

  注意：此时，每个商空间上的向量的加法和乘法都是多重定义的，只有这些定义不冲突时它们才有意义。

• 3.87 定理 商空间是向量空间

  在3.86定义的加法和乘法下商空间是向量空间。

  当然是先证明3.86的Definition是proper的

  设 \(v + U = v' + U,\, w + U = w' + U\) 只有 \((v+w) + U = (v'+w') + U\) 时才有意义，证明简单，略。 数乘同理。

  书上又把证明给省略了，这此按定义逐个验证一下。

  以下用 \(\overline{v}\) 代表 \(v + U\)

  向量加法的结合律 \(\overline{u} + (\overline{v} + \overline{w}) = (\overline{u} + \overline{v}) + \overline{w}\) 成立，都为 \(\overline{u+v+w}\)

  向量加法的交换律 \(\overline{u} + \overline{v} = \overline{v} + \overline{u}\) 显然成立，都为 \(\overline{u+v}\)

  向量加法的单位元 即为 \(\overline{0}\)

  向量加法的逆元素 即为 \(\overline{-v}\) 对任意v ∈ V都存在其逆元素−v ∈ V使得v + (−v) = 0

  标量乘法与标量的域乘法相容 \(a(b \overline{v}) = (ab) \overline{v}, a,b ∈ \mathbb{F}\) 成立，都为 \(\overline{abv}\)

  标量乘法的单位元 域F存在乘法单位元1满足1v = v，即为原始域 \(\mathbb{F}\) 的乘法单位元

  标量乘法对向量加法的分配律 \(a(\overline{u} + \overline{v}) = \overline{au} + \overline{av}\)，成立，都为 \(\overline{a(u+v)}\)

  标量乘法对域加法的分配律 \((a + b) \overline{v} = a \overline{v} + b \overline{v}\)，成立，都为 \(\overline{(a+b)v}\)

• 3.88 定义 quotient map, \(\pi\)

  Suppose \(U\) is a subspace of \(V\). The quotient map \(\pi\) is the linear map \(\pi : V → V / U\) defined by

  \[\pi(v) = v + U,\, v \in V\]

  证明 \(\pi\) 是一个 linear map 作者又略了，比较显然，略。

• 3.89 定理 Dimension of a quotient space

  Suppose \(V\) is finite-dimensional and \(U\) is a subspace of \(V\). Then

  \[\text{dim }V/U = \text{dim }V - \text{dim }U\]

  其实这个比较显然，引入3.88的\(\pi\)之后就可以用 Fundamental Theorem of Linear Maps 证了。

• 3.90 Definition \(\widetilde{T}\)

  Suppose \(T \in \mathcal{L}(V,W)\). Define \(\widetilde{T} : V / (\text{null } T) → W\) by

  \[\widetilde{T}(v + \text{null } T) = T(v)\]

  这个主意很好啊！把T在原空间中相差属于null space的向量归成一个等价类，这样 \(\widetilde{T}\) 还是双射。

• 3.91 Null space and range of \(\widetilde{T}\)

  \(\widetilde{T}\) is a linear map.

  \(\widetilde{T}\) is injective.

  \(\text{range } \widetilde{T} = \text{range } T\).

  \(V / (\text{null } T)\) is isomorphic to \(\text{range } T\).

  证明略。

Duality

• 3.92 定义 linear functional

  A linear functional on V is a linear map from V to F. In other words, a linear functional is an element of \(\mathcal{L}(V,F)\).

• 3.94 定义 dual space, \(V'\)

  The dual space of \(V\), denoted \(V'\), is the vector space of all linear functionals on \(V\). In other words, \(V' = \mathcal{L}(V,F)\)..

• 3.95 定义 对偶空间的维度和原空间相等

  Suppose \(V\) is finite-dimensional. Then V is also finite-dimensional and \(\text{dim }V' = \text{dim}(V)\).

  由3.61直接可得。

• 3.96 定义 Dual basis

  If \(v_1, \cdots, v_n\) is a basis of \(V\), then the dual basis is the list \(\varphi_1, \ldots, \varphi_n\) of elements of \(V'\), where each \(\varphi_j\) is the linear functional on \(V\) such that

  \[\varphi_j (v_j) = 1 \\ \varphi_j (v_\_) = 0\]

• 3.98 Dual basis is a basis of the dual space

  Suppose V is finite-dimensional. Then the dual basis of a basis of V is a basis of V’.

  TODO: 注意这里限定了 \(V\) 是有限维的，还要ponder一下为什么。

  证明见书

• 3.99 定义 Dual map, \(T'\)

  If \(T \in \mathcal{L}(V,W),\) then the dual map of \(T\) is the linear map \(T' \in \mathcal{L}(W',V')\) defined by \(T'(\varphi) = \varphi \circ T\) for \(\varphi \in W'\).

  这里有点看出端倪了，这是要搞转置矩阵啊，TODO: 但是为什么要在对偶空间上搞，对偶空间的主意是怎么想到的，intuitive在哪？

• 3.101 对偶映射的代数性质

  • \((S + T)' = S' + T'\) for all \(S, T \in \mathcal{L}(V,W)\).
  • \((\lambda T)' = \lambda T'\) for all \(\lambda \in \mathbb{F}\) and all \(T \in \mathcal{L}(V,W)\).
  • \((ST)' = T'S'\) for all \(T \in \mathcal{L}(U,V)\) and all \(S \in \mathcal{L}(V,W)\).

  证：前两个都直接由对偶空间的线性性可证，第三个：

  \[(ST)'(\varphi) = \varphi \circ (ST) = (\varphi \circ S) \circ T = T'(S'(\varphi)) = (T'S')(\varphi)\]

The Null Space and Range of the Dual of a Linear Map

Our goal in this subsection is to describe null T’ and range T’ in terms of range T and null T. To do this, we will need the following definition.

• 3.102 定义 annihilator, \(U^0\)

  For \(U \subset V\), the annihilator of \(U\), denoted \(U^0\), is defined by

  \[U^0 = \left\{ \varphi \in V' : \varphi(u) = 0 \text{ for all } u \in U \right\}\]

  尽管这个定义很简明，但是此处比较抽象，还要用自己的话重新说一遍

  \(U^0\) 是一个线性函数(\(V→\mathbb{F}\))的集合，这些线性函数都会把\(U\)中的元素映射成\(\boldsymbol{0}\).

• 3.105 annihilator is a subspace

  Suppose \(U \subset V\). Then \(U^0\) is a subspace of \(V'\).

  证明很短，见书。

• 3.106 Dimension of the annihilator

  Suppose \(V\) is finite-dimensional and \(U\) is a subspace of \(V\). Then

  \[\text{ dim } U + \text{ dim } U^0 = \text{ dim } V\]

  这个定理很直观，但是证明不是很直观。

  证：令\(i \in \mathcal{L}(U,V),\, i(\boldsymbol{u}) = (\boldsymbol{u}) \text{ for } \boldsymbol{u} \in U\)，则

  \[i' = (\backslash f \in V' → i \circ f) \in \mathcal{L}(V',U')\]

  由线性代数基本定理

  \[\text{dim } \text{range}i' + \text{dim }\text{null}i' = \text{dim }V'\]

  由 \(\text{null }i'\) 的定义和 \(U^0\) 的定义可以看出这两个定义相同，

  另一个方向，从U到V再用V→𝔽，不管V的维度比U大还是小，都能等价于从U→𝔽

  若 \(\varphi \in U'\)，则 \(\varphi\) 可以扩展成 \(\psi \in V'\)，所以 \(\varphi = i'(\psi)\)，所以 \(U' \subset \text{range }i'\)，又 \(\text{range }i' \subset U'\)

• 3.107 The null space of \(T'\)

  Suppose \(V\) and \(W\) are finite-dimensional and \(T \in \mathcal{L}(V,W).\) Then

  • \(\text{null } T' = (\text{range } T)^0\).
  • \(\text{dim }\text{ null } T' = \text{dim } \text{null } T + \text{dim } W - \text{dim } V\).

  证：

  \(\begin {aligned} \text{null }T' &= \left\{ f \in W' \middle| f \circ T = \boldsymbol{0} \in V' \right\} \\ \left( \text{range }T \right)^0 &= \left\{ f \in W' \middle| f (x) = 0 \text{ for } x \in \text{range }T \right\} \\ \left( \text{range }T \right)^0 &= \left\{ f \in W' \middle| f (T (v)) = 0 \text{ for } v \in V \right\} \\ \left( \text{range }T \right)^0 &= \left\{ f \in W' \middle| f \circ T = \boldsymbol{0} \in V' \right\} \end {aligned}\)

  \(\begin{aligned} \text{dim } \text{null } T' &= \text{dim } (\text{range} T)^0 \\ &= \text{dim }W - \text{dim }\text{range} T \\ &= \text{dim }W - ( \text{dim }V - \text{dim } \text{null } T ) \\ &= \text{dim } \text{null } T + \text{dim }W - \text{dim }V \end{aligned}\)

• 3.108 T surjective 等价于 T’ injective

  Suppose \(V\) and \(W\) are finite-dimensional and \(T \in \mathcal{L}(V,W)\). Then \(T\) is surjective if and only if \(T'\) is injective.

  Proof The map \(T \in \mathcal{L}(V,W)\) is surjective if and only if \(\text{ range }T=W\), which happens if and only if \((range T)^0 = \{0\}\), which happens if and only if \(\text{null } T' = \{0\}\) [by 3.107(a)], which happens if and only if \(T'\) is injective.

• 3.109 The range of T’

  Suppose \(V\) and \(W\) are finite-dimensional and \(T \in \mathcal{L}(V,W)\). Then

  • \(\text{dim } \text{range } T' = \text{dim }\text{range }T\).
  • \(\text{range }T' = (\text{null } T)^0\).

  证

  \[\begin{aligned} \text{dim }\text{range }T' &= \text{dim }W' - \text{dim }\text{null }T' \\ &= \text{dim }W - \text{dim }(\text{range }T)^0 \\ &= \text{dim }\text{range }T \end{aligned}\]

  First suppose \(\varphi \in \text{range } T'\). Thus there exists \(\psi \in W'\) such that \(\varphi = T'(\psi)\). If \(v \in \text{null } T\), then

  \[\varphi(v) = \left( T'(\psi) \right) v = \left( \psi \circ T \right) (v) = \psi(Tv) = \psi(0) = 0\]

  又

  \begin{aligned} \text{dim }\text{range }T’ &= \text{dim }\text{range }T &= \text{dim }V - \text{dim }\text{null }T &= \text{dim } \left( \text{null }T \right) ^0 \end{aligned}

  由3.69有限维时单双满射等价，证毕

  TODO: 补个图 (这儿可能很不直观，稍后我补个图)

• 3.110 T injective is equivalent to T’ surjective

  Suppose \(V\) and \(W\) are finite-dimensional and \(T \in \mathcal{L}(V,W)\). Then \(T'\) is injective if and only if \(T\) is surjective.

  注：此是对偶对了一半，从T到T’还没有从T’到T，四个子空间说清楚了两个(和相应的对偶空间)

• 3.111 定义，transpose

  对偶了半天终于开始转置了。

• 3.113 转置积

  \[(AC)^T = C^TA^T\]

• 3.114

  Suppose \(T \in \mathcal{L}(V,W)\). Then \(\mathcal{M}(T') = \left( \mathcal{M}(T) \right)^T\)

  书上给了一种证法，但是我觉得此处记号非常混乱，写我自己的证法了。

  注意到对偶空间\(V'\) 是一个 \(V \times \mathbb{F}\)，又是线性映射，说白了就是点乘么。所以 \(V'\) 是的向量，设为 \(\boldsymbol{u}\)， 是把\(V\)变成\(\mathbb{F}\)，就是\(\boldsymbol{u} = \boldsymbol{u} \cdot \boldsymbol{v}\)么(等号前u为V’的元素，等号后u为等号前u在标准基下的一组表示)，

  所以，设 \(\varphi = \mathcal{M}(T') \psi,\,(\varphi \in V', \psi \in W')\)，由 \(T'\) 的定义，

  \[\varphi (v) = \psi ( \mathcal{M}(T)v )\]

  就是说

  \[\varphi^T = \psi^T \mathcal{M}(T) \\ \varphi = \left( \mathcal{M}(T) \right)^T \psi\]

  所以 \(\left( \mathcal{M}(T) \right)^T = \mathcal{M}(T')\)

• 3.115 定义 行秩 列秩 (m-by-n)

  矩阵的行秩是矩阵的行空间的秩 矩阵的列秩是矩阵的列空间的秩

• 3.117 线性变换的值空间维度等于相应矩阵的列秩。

• 3.118 行秩等于列秩

  见 3.109 的图

• 3.119 定义 秩

  既然行秩等于列秩，就定义它们为秩好了。

  注：此时第三章abruptly讲完了。。。

后续在这里。。。