⭐ 线性空间的8条定义公理（4条加法 + 4条数乘）
⭐ 线性子空间的判定条件：非空（或 0 ∈ S 0\in S 0 ∈ S
⭐ 基、维数、坐标的定义与唯一性
⭐ 同构映射的定义及其保持的性质
⭐ 过渡矩阵的定义与性质（可逆）
⭐ 坐标变换公式：x = C y x = Cy x = C y y = C − 1 x y = C^{-1}x y = C − 1 x
⭐ 线性变换的定义（可加性 + 齐次性）
⭐ 线性变换的矩阵表示及其唯一性
⭐ 线性变换在不同基下的矩阵相似：B = C − 1 A C B = C^{-1}AC B = C − 1 A C
⭐ 秩-零度定理：dim ( Im ( σ ) ) + dim ( Ker ( σ ) ) = dim ( V ) \dim(\text{Im}(\sigma)) + \dim(\text{Ker}(\sigma)) = \dim(V) dim ( Im ( σ )) + dim ( Ker ( σ )) = dim ( V )
常见反例：全体“次数恰为 n n n b ≠ 0 b\neq 0 b = 0
数域 ：设 P P P P P P 0 0 0 1 1 1 a , b ∈ P a,b\in P a , b ∈ P b ≠ 0 b\neq 0 b = 0 a / b ∈ P a/b\in P a / b ∈ P P P P
常见数域 ：全体有理数 Q \mathbb{Q} Q R \mathbb{R} R C \mathbb{C} C
例 ：P = { a + b 2 ∣ a , b ∈ Q } P = \{a + b\sqrt{2} \mid a, b \in \mathbb{Q}\} P = { a + b 2 ∣ a , b ∈ Q }
线性空间 ：设 V V V P P P V V V V V V P P P 线性空间 。[来源：p3-4]
加法公理（4条） ：
加法交换律 ：α + β = β + α \alpha + \beta = \beta + \alpha α + β = β + α 加法结合律 ：( α + β ) + γ = α + ( β + γ ) (\alpha + \beta) + \gamma = \alpha + (\beta + \gamma) ( α + β ) + γ = α + ( β + γ ) 零元素存在 ：在 V V V 0 0 0 零元素 ，对所定义的加法满足：对任意 α ∈ V \alpha \in V α ∈ V α + 0 = α \alpha + 0 = \alpha α + 0 = α 负元素存在 ：对任意 α ∈ V \alpha \in V α ∈ V β ∈ V \beta \in V β ∈ V α + β = 0 \alpha + \beta = 0 α + β = 0 β = − α \beta = -\alpha β = − α β \beta β α \alpha α 负元素 [来源：p3]
数乘公理（4条） ：
5. 数乘单位元 ：对任意 α ∈ V \alpha \in V α ∈ V 1 ⋅ α = α 1 \cdot \alpha = \alpha 1 ⋅ α = α 数乘结合律 ：对任意 α ∈ V \alpha \in V α ∈ V k , l ∈ P k, l \in P k , l ∈ P k ⋅ ( l ⋅ α ) = ( k l ) ⋅ α k \cdot (l \cdot \alpha) = (kl) \cdot \alpha k ⋅ ( l ⋅ α ) = ( k l ) ⋅ α 数乘对向量加法的分配律 ：对任意 α , β ∈ V \alpha, \beta \in V α , β ∈ V k ∈ P k \in P k ∈ P k ( α + β ) = k ⋅ α + k ⋅ β k(\alpha + \beta) = k \cdot \alpha + k \cdot \beta k ( α + β ) = k ⋅ α + k ⋅ β 数乘对数加法的分配律 ：对任意 α ∈ V \alpha \in V α ∈ V k , l ∈ P k, l \in P k , l ∈ P ( k + l ) α = k ⋅ α + l ⋅ α (k + l)\alpha = k \cdot \alpha + l \cdot \alpha ( k + l ) α = k ⋅ α + l ⋅ α
线性空间的性质 ：[来源：p12]
零元素是唯一的
每个元素的负元素是唯一的
0 ⋅ α = 0 0 \cdot \alpha = 0 0 ⋅ α = 0 ( − 1 ) ⋅ α = − α (-1) \cdot \alpha = -\alpha ( − 1 ) ⋅ α = − α k ⋅ 0 = 0 k \cdot 0 = 0 k ⋅ 0 = 0 ⭐ 若 k ⋅ α = 0 k \cdot \alpha = 0 k ⋅ α = 0 k = 0 k = 0 k = 0 α = 0 \alpha = 0 α = 0
全体 n n n V V V R \mathbb{R} R R n \mathbb{R}^n R n
全体 m × n m \times n m × n V V V R \mathbb{R} R R m × n \mathbb{R}^{m \times n} R m × n
全体次数不超过 n − 1 n-1 n − 1 x x x V V V R \mathbb{R} R R [ x ] n \mathbb{R}[x]_n R [ x ] n
闭区间 [ a , b ] [a, b] [ a , b ] V V V R \mathbb{R} R C [ a , b ] C[a,b] C [ a , b ]
⭐ 实系数的齐次线性方程组的解的全体 的集合 V V V
仅由 n n n V V V
例 ：令 V = R + V = \mathbb{R}^+ V = R + P = R P = \mathbb{R} P = R a ⊕ b = def a ⋅ b a \oplus b \stackrel{\text{def}}{=} a \cdot b a ⊕ b = def a ⋅ b k ∘ a = def a k k \circ a \stackrel{\text{def}}{=} a^k k ∘ a = def a k R + \mathbb{R}^+ R + R \mathbb{R} R
加法运算的零元素为 1 1 1 a ⊕ 1 = a ⋅ 1 = a a \oplus 1 = a \cdot 1 = a a ⊕ 1 = a ⋅ 1 = a
加法运算的负元素：对 ∀ a ∈ R + \forall a \in \mathbb{R}^+ ∀ a ∈ R + 1 a ∈ R + \frac{1}{a} \in \mathbb{R}^+ a 1 ∈ R + a ⊕ 1 a = a ⋅ 1 a = 1 a \oplus \frac{1}{a} = a \cdot \frac{1}{a} = 1 a ⊕ a 1 = a ⋅ a 1 = 1
⭐ 全体“次数恰为 n n n ：例如 x n x^n x n − x n -x^n − x n 0 0 0 n n n
⭐ 全体实数的集合 V V V P = C P = \mathbb{C} P = C ：因为数乘不封闭（复数乘实数可能不再是实数，如 i ⋅ 1 = i ∉ R i \cdot 1 = i \notin \mathbb{R} i ⋅ 1 = i ∈ / R
⭐ 当 b ≠ 0 b\neq 0 b = 0 A x = b Ax = b A x = b ：例如若 x 1 , x 2 x_1,x_2 x 1 , x 2 A ( x 1 + x 2 ) = 2 b ≠ b A(x_1+x_2)=2b\neq b A ( x 1 + x 2 ) = 2 b = b A ( k x 1 ) = k b ≠ b A(kx_1)=kb\neq b A ( k x 1 ) = k b = b
数乘不满足 1 ⋅ α = α 1 \cdot \alpha = \alpha 1 ⋅ α = α ：设 V = R 2 = { ( x 1 , x 2 ) ∣ x i ∈ R , i = 1 , 2 } V = \mathbb{R}^2 = \{(x_1, x_2) \mid x_i \in \mathbb{R}, i = 1, 2\} V = R 2 = {( x 1 , x 2 ) ∣ x i ∈ R , i = 1 , 2 } k ∘ ( x 1 , x 2 ) = ( k x 1 , 0 ) k \circ (x_1, x_2) = (kx_1, 0) k ∘ ( x 1 , x 2 ) = ( k x 1 , 0 ) V V V α = ( x 1 , x 2 ) ∈ V \alpha = (x_1, x_2) \in V α = ( x 1 , x 2 ) ∈ V x 2 ≠ 0 x_2 \neq 0 x 2 = 0 1 ∘ ( x 1 , x 2 ) = ( x 1 , 0 ) ≠ ( x 1 , x 2 ) 1 \circ (x_1, x_2) = (x_1, 0) \neq (x_1, x_2) 1 ∘ ( x 1 , x 2 ) = ( x 1 , 0 ) = ( x 1 , x 2 )
线性子空间 ：设 V V V P P P S ⊆ V S \subseteq V S ⊆ V S ≠ ∅ S \neq \emptyset S = ∅ S S S V V V P P P S S S V V V 线性子空间 （简称子空间 ）。[来源：p13-14]
⭐ 子空间判定定理 ：S ⊆ V S \subseteq V S ⊆ V S ≠ ∅ S \neq \emptyset S = ∅ V V V 当且仅当 S S S V V V
即：∀ α , β ∈ S \forall \alpha, \beta \in S ∀ α , β ∈ S α + β ∈ S \alpha + \beta \in S α + β ∈ S
且：∀ α ∈ S , k ∈ P \forall \alpha \in S, k \in P ∀ α ∈ S , k ∈ P k α ∈ S k\alpha \in S k α ∈ S
设 A A A m × n m \times n m × n A x = 0 Ax = 0 A x = 0 ⊆ R n \subseteq \mathbb{R}^n ⊆ R n
V 1 = { ( 0 , a 2 , a 3 , ⋯ , a n ) ∣ a i ∈ R , i = 2 , 3 , ⋯ , n } ⊆ R n V_1 = \{(0, a_2, a_3, \cdots, a_n) \mid a_i \in \mathbb{R}, i = 2, 3, \cdots, n\} \subseteq \mathbb{R}^n V 1 = {( 0 , a 2 , a 3 , ⋯ , a n ) ∣ a i ∈ R , i = 2 , 3 , ⋯ , n } ⊆ R n V 3 = { A n × n ∣ A n × n ∈ R n × n , A T = A } ⊆ R n × n V_3 = \{A_{n \times n} \mid A_{n \times n} \in \mathbb{R}^{n \times n}, A^T = A\} \subseteq \mathbb{R}^{n \times n} V 3 = { A n × n ∣ A n × n ∈ R n × n , A T = A } ⊆ R n × n
验证子空间的步骤：验证加法和数乘封闭即可。[来源：p16，标注"容易验证：加法和数乘封闭"]
线性空间中的向量 ：线性空间的元素也称为"向量"。可以仿照 n n n
基 ：线性空间 V V V V V V B B B V V V B B B V V V
维数 ：线性空间 V V V dim ( V ) \dim(V) dim ( V ) dim ( V ) \dim(V) dim ( V ) V V V
坐标 ：就是向量在基下的表示系数，具有唯一性 。[来源：p28，红字标注]
向量在基下的坐标是唯一的。[来源：p28]
向量的线性表示可以用矩阵形式表示：若 β 1 , β 2 , ⋯ , β t \beta_1, \beta_2, \cdots, \beta_t β 1 , β 2 , ⋯ , β t α 1 , α 2 , ⋯ , α m \alpha_1, \alpha_2, \cdots, \alpha_m α 1 , α 2 , ⋯ , α m
( β 1 , β 2 , ⋯ , β t ) = ( α 1 , α 2 , ⋯ , α m ) B (\beta_1, \beta_2, \cdots, \beta_t) = (\alpha_1, \alpha_2, \cdots, \alpha_m)B ( β 1 , β 2 , ⋯ , β t ) = ( α 1 , α 2 , ⋯ , α m ) B 其中 B B B m × t m \times t m × t
对同一向量组而言，极大线性无关组一般不唯一，但其秩（即极大线性无关组所含向量的个数）是唯一确定的。[来源：p33]
极大线性无关组固定后，每个向量的表示系数唯一。[来源：p33]
对某向量组 S S S r ( S ) r(S) r ( S ) S S S r ( S ) r(S) r ( S ) S S S
对某向量组 S S S r ( S ) r(S) r ( S ) r ( S ) r(S) r ( S )
若向量组 T T T S S S T T T r ( S ) r(S) r ( S ) T T T S S S ⟨ S ⟩ \langle S\rangle ⟨ S ⟩
R [ x ] n \mathbb{R}[x]_n R [ x ] n n − 1 n-1 n − 1 1 , x , x 2 , ⋯ , x n − 1 1, x, x^2, \cdots, x^{n-1} 1 , x , x 2 , ⋯ , x n − 1 n n n
任意多项式 f ( x ) = a 0 + a 1 x + ⋯ + a n − 1 x n − 1 f(x) = a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} f ( x ) = a 0 + a 1 x + ⋯ + a n − 1 x n − 1 ( a 0 , a 1 , ⋯ , a n − 1 ) T (a_0, a_1, \cdots, a_{n-1})^T ( a 0 , a 1 , ⋯ , a n − 1 ) T
R 2 × 3 \mathbb{R}^{2 \times 3} R 2 × 3 2 × 3 2 \times 3 2 × 3 E 11 , E 12 , E 13 , E 21 , E 22 , E 23 E_{11}, E_{12}, E_{13}, E_{21}, E_{22}, E_{23} E 11 , E 12 , E 13 , E 21 , E 22 , E 23 6 6 6
同构映射（线性同构） ：若映射 ϕ \phi ϕ ϕ \phi ϕ
设 V V V P P P n n n V V V α 1 , α 2 , ⋯ , α n \alpha_1, \alpha_2, \cdots, \alpha_n α 1 , α 2 , ⋯ , α n α ∈ V \alpha \in V α ∈ V α \alpha α α 1 , α 2 , ⋯ , α n \alpha_1, \alpha_2, \cdots, \alpha_n α 1 , α 2 , ⋯ , α n x x x α = ( α 1 , α 2 , ⋯ , α n ) x \alpha = (\alpha_1, \alpha_2, \cdots, \alpha_n)x α = ( α 1 , α 2 , ⋯ , α n ) x x ∈ P n x \in P^n x ∈ P n n n n ϕ : V → P n \phi: V \to P^n ϕ : V → P n α ∈ V \alpha \in V α ∈ V ϕ ( α ) = x \phi(\alpha) = x ϕ ( α ) = x
映射 ϕ : V → P n \phi: V \to P^n ϕ : V → P n
ϕ ( α + β ) = ϕ ( α ) + ϕ ( β ) \phi(\alpha + \beta) = \phi(\alpha) + \phi(\beta) ϕ ( α + β ) = ϕ ( α ) + ϕ ( β ) ϕ ( k α ) = k ϕ ( α ) \phi(k\alpha) = k\phi(\alpha) ϕ ( k α ) = k ϕ ( α )
⭐ 同构把线性相关的向量组变成线性相关的向量组 。[来源：p43，红字标注]
⭐ 同构把线性无关的向量组变成线性无关的向量组 。[来源：p43，红字标注]
同构把零元素变成零元素，负元素变成负元素。[来源：p43，红字标注]
同构把一组基变成一组基。[来源：p43，红字标注]
同构保持向量组的秩不变。[来源：p43，红字标注]
同构保持线性空间的维数不变。[来源：p43，红字标注]
过渡矩阵 ：设 V V V P P P n n n α 1 , α 2 , ⋯ , α n \alpha_1, \alpha_2, \cdots, \alpha_n α 1 , α 2 , ⋯ , α n β 1 , β 2 , ⋯ , β n \beta_1, \beta_2, \cdots, \beta_n β 1 , β 2 , ⋯ , β n β 1 , β 2 , ⋯ , β n \beta_1, \beta_2, \cdots, \beta_n β 1 , β 2 , ⋯ , β n α 1 , α 2 , ⋯ , α n \alpha_1, \alpha_2, \cdots, \alpha_n α 1 , α 2 , ⋯ , α n
( β 1 , β 2 , ⋯ , β n ) = ( α 1 , α 2 , ⋯ , α n ) C (\beta_1, \beta_2, \cdots, \beta_n) = (\alpha_1, \alpha_2, \cdots, \alpha_n)C ( β 1 , β 2 , ⋯ , β n ) = ( α 1 , α 2 , ⋯ , α n ) C 称矩阵 C C C 由基 α 1 , α 2 , ⋯ , α n \alpha_1, \alpha_2, \cdots, \alpha_n α 1 , α 2 , ⋯ , α n β 1 , β 2 , ⋯ , β n \beta_1, \beta_2, \cdots, \beta_n β 1 , β 2 , ⋯ , β n 。[来源：p3，黄色标注]
⭐ 过渡矩阵是可逆矩阵 。[来源：p3]
C − 1 C^{-1} C − 1 β 1 , β 2 , ⋯ , β n \beta_1, \beta_2, \cdots, \beta_n β 1 , β 2 , ⋯ , β n α 1 , α 2 , ⋯ , α n \alpha_1, \alpha_2, \cdots, \alpha_n α 1 , α 2 , ⋯ , α n
( α 1 , α 2 , ⋯ , α n ) = ( β 1 , β 2 , ⋯ , β n ) C − 1 (\alpha_1, \alpha_2, \cdots, \alpha_n) = (\beta_1, \beta_2, \cdots, \beta_n)C^{-1} ( α 1 , α 2 , ⋯ , α n ) = ( β 1 , β 2 , ⋯ , β n ) C − 1 [来源：p3]
设 ξ ∈ V \xi \in V ξ ∈ V V V V ξ \xi ξ ( α 1 , α 2 , ⋯ , α n ) (\alpha_1, \alpha_2, \cdots, \alpha_n) ( α 1 , α 2 , ⋯ , α n ) ( x 1 , x 2 , ⋯ , x n ) T (x_1, x_2, \cdots, x_n)^T ( x 1 , x 2 , ⋯ , x n ) T ξ = ( α 1 , α 2 , ⋯ , α n ) x \xi = (\alpha_1, \alpha_2, \cdots, \alpha_n)x ξ = ( α 1 , α 2 , ⋯ , α n ) x ξ \xi ξ ( β 1 , β 2 , ⋯ , β n ) (\beta_1, \beta_2, \cdots, \beta_n) ( β 1 , β 2 , ⋯ , β n ) ( y 1 , y 2 , ⋯ , y n ) T (y_1, y_2, \cdots, y_n)^T ( y 1 , y 2 , ⋯ , y n ) T ξ = ( β 1 , β 2 , ⋯ , β n ) y \xi = (\beta_1, \beta_2, \cdots, \beta_n)y ξ = ( β 1 , β 2 , ⋯ , β n ) y
⭐ 坐标变换公式 ：若 ( β 1 , β 2 , ⋯ , β n ) = ( α 1 , α 2 , ⋯ , α n ) C (\beta_1, \beta_2, \cdots, \beta_n) = (\alpha_1, \alpha_2, \cdots, \alpha_n)C ( β 1 , β 2 , ⋯ , β n ) = ( α 1 , α 2 , ⋯ , α n ) C
x = C y 或者 y = C − 1 x x = Cy \quad \text{或者} \quad y = C^{-1}x x = C y 或者 y = C − 1 x 其中 x x x y y y ( β 1 , ⋯ , β n ) = ( α 1 , ⋯ , α n ) C (\beta_1, \cdots, \beta_n) = (\alpha_1, \cdots, \alpha_n)C ( β 1 , ⋯ , β n ) = ( α 1 , ⋯ , α n ) C C C C C C C x = C y x=Cy x = C y C − 1 C^{-1} C − 1 y = C − 1 x y=C^{-1}x y = C − 1 x
新基向量在旧基下的坐标 ：β j \beta_j β j α 1 , α 2 , ⋯ , α n \alpha_1, \alpha_2, \cdots, \alpha_n α 1 , α 2 , ⋯ , α n C C C j j j
旧基向量在新基下的坐标 ：由于 ( α 1 , α 2 , ⋯ , α n ) = ( β 1 , β 2 , ⋯ , β n ) C − 1 (\alpha_1, \alpha_2, \cdots, \alpha_n) = (\beta_1, \beta_2, \cdots, \beta_n)C^{-1} ( α 1 , α 2 , ⋯ , α n ) = ( β 1 , β 2 , ⋯ , β n ) C − 1 α j \alpha_j α j β 1 , β 2 , ⋯ , β n \beta_1, \beta_2, \cdots, \beta_n β 1 , β 2 , ⋯ , β n C − 1 C^{-1} C − 1 j j j
求在两组基下有相同坐标的向量 ：设所求向量 α \alpha α x x x x = C x x = Cx x = C x ( E − C ) x = 0 (E - C)x = 0 ( E − C ) x = 0 ξ 1 , ξ 2 , ⋯ \xi_1, \xi_2, \cdots ξ 1 , ξ 2 , ⋯ α = ( α 1 , α 2 , α 3 ) ( k 1 ξ 1 + k 2 ξ 2 + ⋯ ) \alpha = (\alpha_1, \alpha_2, \alpha_3)(k_1\xi_1 + k_2\xi_2 + \cdots) α = ( α 1 , α 2 , α 3 ) ( k 1 ξ 1 + k 2 ξ 2 + ⋯ )
⭐ 利用标准基作为桥梁 ：E 11 , E 12 , E 21 , E 22 E_{11}, E_{12}, E_{21}, E_{22} E 11 , E 12 , E 21 , E 22
设 E 11 , E 12 , E 21 , E 22 E_{11}, E_{12}, E_{21}, E_{22} E 11 , E 12 , E 21 , E 22 A 1 , A 2 , A 3 , A 4 A_1, A_2, A_3, A_4 A 1 , A 2 , A 3 , A 4 C C C
设 E 11 , E 12 , E 21 , E 22 E_{11}, E_{12}, E_{21}, E_{22} E 11 , E 12 , E 21 , E 22 B 1 , B 2 , B 3 , B 4 B_1, B_2, B_3, B_4 B 1 , B 2 , B 3 , B 4 D D D
则 ( B 1 , B 2 , B 3 , B 4 ) = ( A 1 , A 2 , A 3 , A 4 ) C − 1 D (B_1, B_2, B_3, B_4) = (A_1, A_2, A_3, A_4)C^{-1}D ( B 1 , B 2 , B 3 , B 4 ) = ( A 1 , A 2 , A 3 , A 4 ) C − 1 D
变换 ：设 V V V P P P V V V σ \sigma σ 对任意的 α ∈ V \alpha \in V α ∈ V σ ( α ) ∈ V \sigma(\alpha) \in V σ ( α ) ∈ V α \alpha α ，则称 σ \sigma σ V V V 变换 。σ ( α ) \sigma(\alpha) σ ( α ) α \alpha α σ \sigma σ 像 ，α \alpha α σ ( α ) \sigma(\alpha) σ ( α ) 一个原像 。[来源：p2，红字标注]
⭐ 线性变换 ：设 V V V P P P σ \sigma σ V V V α , β ∈ V \alpha, \beta \in V α , β ∈ V k ∈ P k \in P k ∈ P
σ ( α + β ) = σ ( α ) + σ ( β ) \sigma(\alpha + \beta) = \sigma(\alpha) + \sigma(\beta) σ ( α + β ) = σ ( α ) + σ ( β ) 可加性 ）σ ( k α ) = k σ ( α ) \sigma(k\alpha) = k\sigma(\alpha) σ ( k α ) = k σ ( α ) 齐次性 ）
则称 σ \sigma σ V V V 线性变换 。
⭐ 线性变换 σ \sigma σ V V V α 1 , α 2 , ⋯ , α m \alpha_1, \alpha_2, \cdots, \alpha_m α 1 , α 2 , ⋯ , α m σ ( α 1 ) , σ ( α 2 ) , ⋯ , σ ( α m ) \sigma(\alpha_1), \sigma(\alpha_2), \cdots, \sigma(\alpha_m) σ ( α 1 ) , σ ( α 2 ) , ⋯ , σ ( α m ) 。[来源：p5]
【整理自PPT】注意 ：线性变换不一定把线性无关的向量组变成线性无关的向量组。例如零变换 σ ( α ) = 0 \sigma(\alpha) = 0 σ ( α ) = 0
⭐ 线性变换在基下的矩阵 ：设 n n n V V V α 1 , α 2 , ⋯ , α n \alpha_1, \alpha_2, \cdots, \alpha_n α 1 , α 2 , ⋯ , α n σ \sigma σ
( σ ( α 1 ) , σ ( α 2 ) , ⋯ , σ ( α n ) ) = ( α 1 , α 2 , ⋯ , α n ) A (\sigma(\alpha_1), \sigma(\alpha_2), \cdots, \sigma(\alpha_n)) = (\alpha_1, \alpha_2, \cdots, \alpha_n)A ( σ ( α 1 ) , σ ( α 2 ) , ⋯ , σ ( α n )) = ( α 1 , α 2 , ⋯ , α n ) A 称矩阵 A A A 线性变换 σ \sigma σ α 1 , α 2 , ⋯ , α n \alpha_1, \alpha_2, \cdots, \alpha_n α 1 , α 2 , ⋯ , α n 。
⭐ 只要确定了线性变换 σ \sigma σ σ \sigma σ 。[来源：p10，蓝字标注]
线性变换在基下的矩阵是唯一的 ：由于 σ ( α j ) \sigma(\alpha_j) σ ( α j ) α 1 , α 2 , ⋯ , α n \alpha_1, \alpha_2, \cdots, \alpha_n α 1 , α 2 , ⋯ , α n σ \sigma σ α 1 , α 2 , ⋯ , α n \alpha_1, \alpha_2, \cdots, \alpha_n α 1 , α 2 , ⋯ , α n
定理（存在性与唯一性） ：设 α 1 , α 2 , ⋯ , α n \alpha_1, \alpha_2, \cdots, \alpha_n α 1 , α 2 , ⋯ , α n P P P n n n V V V A A A n n n 唯一的 线性变换 σ \sigma σ α 1 , α 2 , ⋯ , α n \alpha_1, \alpha_2, \cdots, \alpha_n α 1 , α 2 , ⋯ , α n A A A
⭐ 一一对应关系 ：设 α 1 , α 2 , ⋯ , α n \alpha_1, \alpha_2, \cdots, \alpha_n α 1 , α 2 , ⋯ , α n n n n V V V n n n ( σ ( α 1 ) , σ ( α 2 ) , ⋯ , σ ( α n ) ) = ( α 1 , α 2 , ⋯ , α n ) A (\sigma(\alpha_1), \sigma(\alpha_2), \cdots, \sigma(\alpha_n)) = (\alpha_1, \alpha_2, \cdots, \alpha_n)A ( σ ( α 1 ) , σ ( α 2 ) , ⋯ , σ ( α n )) = ( α 1 , α 2 , ⋯ , α n ) A
⭐ 坐标变换定理 ：设 α 1 , α 2 , ⋯ , α n \alpha_1, \alpha_2, \cdots, \alpha_n α 1 , α 2 , ⋯ , α n P P P n n n V V V V V V σ \sigma σ A A A 若向量 α \alpha α x x x α \alpha α σ ( α ) \sigma(\alpha) σ ( α ) A x Ax A x 。[来源：p34，红字标注]
⭐ 定理 ：设 α 1 , α 2 , ⋯ , α n \alpha_1, \alpha_2, \cdots, \alpha_n α 1 , α 2 , ⋯ , α n β 1 , β 2 , ⋯ , β n \beta_1, \beta_2, \cdots, \beta_n β 1 , β 2 , ⋯ , β n P P P n n n V V V V V V σ \sigma σ A A A B B B α 1 , α 2 , ⋯ , α n \alpha_1, \alpha_2, \cdots, \alpha_n α 1 , α 2 , ⋯ , α n β 1 , β 2 , ⋯ , β n \beta_1, \beta_2, \cdots, \beta_n β 1 , β 2 , ⋯ , β n C C C
B = C − 1 A C B = C^{-1}AC B = C − 1 A C 即矩阵 B B B A A A 。[来源：p36，红字标注]
⭐ 核心结论 ：线性变换在不同基下的矩阵是相似的 。[来源：p37，红字标注]
【整理自PPT】逆命题 ：若两个矩阵 A A A B B B A A A B B B
值域 ：Im ( σ ) = { σ ( α ) ∣ α ∈ V } \text{Im}(\sigma) = \{\sigma(\alpha) \mid \alpha \in V\} Im ( σ ) = { σ ( α ) ∣ α ∈ V } σ \sigma σ 值域 ，记作 Im ( σ ) \text{Im}(\sigma) Im ( σ )
核 ：Ker ( σ ) = { α ∈ V ∣ σ ( α ) = 0 } \text{Ker}(\sigma) = \{\alpha \in V \mid \sigma(\alpha) = 0\} Ker ( σ ) = { α ∈ V ∣ σ ( α ) = 0 } 核 ，记作 Ker ( σ ) \text{Ker}(\sigma) Ker ( σ )
秩 ：线性变换 σ \sigma σ σ \sigma σ 秩 ，记作 r ( σ ) r(\sigma) r ( σ )
零度 ：线性变换 σ \sigma σ σ \sigma σ 零度 ，可记作 dim ( Ker ( σ ) ) \dim(\text{Ker}(\sigma)) dim ( Ker ( σ )) r ( σ − 1 ( 0 ) ) r(\sigma^{-1}(0)) r ( σ − 1 ( 0 ))
dim ( Im ( σ ) ) + dim ( Ker ( σ ) ) = dim ( V ) = n \dim(\text{Im}(\sigma)) + \dim(\text{Ker}(\sigma)) = \dim(V) = n dim ( Im ( σ )) + dim ( Ker ( σ )) = dim ( V ) = n 即线性变换 σ \sigma σ σ \sigma σ V V V 。
公式/结论 说明 α + 0 = α \alpha + 0 = \alpha α + 0 = α 零元素 α + ( − α ) = 0 \alpha + (-\alpha) = 0 α + ( − α ) = 0 负元素 1 ⋅ α = α 1 \cdot \alpha = \alpha 1 ⋅ α = α 数乘单位元 0 ⋅ α = 0 0 \cdot \alpha = 0 0 ⋅ α = 0 数0乘向量 k ⋅ 0 = 0 k \cdot 0 = 0 k ⋅ 0 = 0 数乘零向量 ( − 1 ) ⋅ α = − α (-1) \cdot \alpha = -\alpha ( − 1 ) ⋅ α = − α 负元素的另一表达 k ⋅ α = 0 ⇒ k = 0 k \cdot \alpha = 0 \Rightarrow k = 0 k ⋅ α = 0 ⇒ k = 0 α = 0 \alpha = 0 α = 0 零因子性质
公式 说明 ( β 1 , ⋯ , β n ) = ( α 1 , ⋯ , α n ) C (\beta_1, \cdots, \beta_n) = (\alpha_1, \cdots, \alpha_n)C ( β 1 , ⋯ , β n ) = ( α 1 , ⋯ , α n ) C 过渡矩阵定义 ( α 1 , ⋯ , α n ) = ( β 1 , ⋯ , β n ) C − 1 (\alpha_1, \cdots, \alpha_n) = (\beta_1, \cdots, \beta_n)C^{-1} ( α 1 , ⋯ , α n ) = ( β 1 , ⋯ , β n ) C − 1 逆过渡矩阵 x = C y x = Cy x = C y 坐标变换（旧基坐标 = 过渡矩阵 × 新基坐标） y = C − 1 x y = C^{-1}x y = C − 1 x 坐标变换的逆形式
公式 说明 ( σ ( α 1 ) , ⋯ , σ ( α n ) ) = ( α 1 , ⋯ , α n ) A (\sigma(\alpha_1), \cdots, \sigma(\alpha_n)) = (\alpha_1, \cdots, \alpha_n)A ( σ ( α 1 ) , ⋯ , σ ( α n )) = ( α 1 , ⋯ , α n ) A 线性变换的矩阵定义 若在同一组基下 α \alpha α x x x σ ( α ) \sigma(\alpha) σ ( α ) A x Ax A x 坐标与矩阵均相对于同一组基 B = C − 1 A C B = C^{-1}AC B = C − 1 A C 不同基下矩阵的相似关系 dim ( Im ( σ ) ) + dim ( Ker ( σ ) ) = n \dim(\text{Im}(\sigma)) + \dim(\text{Ker}(\sigma)) = n dim ( Im ( σ )) + dim ( Ker ( σ )) = n 秩-零度定理
本知识集锦严格基于上海交通大学线性代数课程第35、36、38讲PPT内容整理