非空集合V，數域F，對加法和數乘封閉，即∀ α , β ∈ V \forall \alpha,\ \beta \in V ∀ α , β ∈ V α + β ∈ V \alpha+\beta \in V α + β ∈ V ∀ k ∈ F ， α ∈ V \forall k \in F，\alpha \in V ∀ k ∈ F ， α ∈ V k α ∈ V k\alpha \in V k α ∈ V α + β = β + α \alpha + \beta = \beta + \alpha α + β = β + α ( α + β ) + γ = α + ( β + γ ) (\alpha + \beta) + \gamma = \alpha + (\beta + \gamma) ( α + β ) + γ = α + ( β + γ ) ∃ α 0 ∈ V , ∀ α ∈ V , α + α 0 = α , 記 α 0 = 0 \exist \alpha_0 \in V,\ \forall \alpha \in V,\ \alpha + \alpha_0 = \alpha,\ 記\alpha_0 = 0 ∃ α 0 ∈ V , ∀ α ∈ V , α + α 0 = α , 記 α 0 = 0 ∀ α ∈ V , ∃ β ∈ V , 使 α + β = 0 , 記 β = − α \forall \alpha \in V,\ \exist \beta \in V,\ 使\alpha + \beta = 0,\ 記\beta = -\alpha ∀ α ∈ V , ∃ β ∈ V , 使 α + β = 0 , 記 β = − α ∃ 1 ∈ F \exist 1 \in F ∃ 1 ∈ F 1 ⋅ α = α 1 \cdot \alpha = \alpha 1 ⋅ α = α ( k l ) α = k ( l α ) (kl) \alpha = k(l \alpha) ( k l ) α = k ( l α ) k ( α + β ) = k α + k β k(\alpha + \beta) = k \alpha + k \beta k ( α + β ) = k α + k β ( k + l ) α = k α + l α (k + l) \alpha = k \alpha + l \alpha ( k + l ) α = k α + l α V是數域F上的線性空間。V中元素稱爲 向量 。F爲實(復)數域 時，稱V爲實(復)線性空間 。
V中零元素唯一
V中任一元素的負元素唯一
設0爲數0，0 ⃗ \vec{0} 0 0 ⋅ α = 0 ⃗ 0 \cdot \alpha = \vec{0} 0 ⋅ α = 0 k ⋅ 0 ⃗ = 0 ⃗ , k ∈ F k \cdot \vec{0} = \vec{0},\ k \in F k ⋅ 0 = 0 , k ∈ F k ⋅ α = 0 ⃗ k \cdot \alpha = \vec{0} k ⋅ α = 0 k = 0 k=0 k = 0 α = 0 ⃗ \alpha = \vec{0} α = 0 ( − 1 ) α = − α (-1) \alpha = - \alpha ( − 1 ) α = − α
基： 線性空間V中，若存在一組線性無關的向量α 1 , α 2 , . . . , α n \alpha_1,\ \alpha_2,\ ...,\ \alpha_n α 1 , α 2 , . . . , α n { α 1 , α 2 , . . . , α n } \{\alpha_1,\ \alpha_2,\ ...,\ \alpha_n\} { α 1 , α 2 , . . . , α n } 維數： 基中所含向量個數，記爲d i m V = n dimV = n d i m V = n 基就是向量集合V中的極大線性無關組，因此線性空間的基不唯一 標準正交基： 內積空間[ V n ( F ) ; ( α , β ) ] [V_n(F);\ (\alpha,\ \beta)] [ V n ( F ) ; ( α , β ) ] { ϵ 1 , ϵ 2 , ⋯ , ϵ m } \{\epsilon_1,\ \epsilon_2,\ \cdots,\ \epsilon_m\} { ϵ 1 , ϵ 2 , ⋯ , ϵ m } ( ϵ i , ϵ j ) = { 1 i = j 0 i ≠ j (\epsilon_i,\ \epsilon_j) =
\begin{cases}
1 & i = j \\
0 & i \neq j
\end{cases} ( ϵ i , ϵ j ) = { 1 0 i = j i = j
若{ E i j , i = 1 , 2 , . . . , m ; j = 1 , 2 , . . . , n } \{E_{ij},\ i=1,\ 2,\ ...,\ m;\ j=1,\ 2,\ ...,\ n \} { E i j , i = 1 , 2 , . . . , m ; j = 1 , 2 , . . . , n } R m ∗ n R^{m*n} R m ∗ n d i m R m ∗ n = m ∗ n dim\ R^{m*n} = m * n d i m R m ∗ n = m ∗ n
設α 1 , α 2 , ⋯ , α n \alpha_1,\ \alpha_2,\ \cdots,\ \alpha_n α 1 , α 2 , ⋯ , α n V n ( F ) V_n(F) V n ( F ) ∀ β ∈ V \forall \beta \in V ∀ β ∈ V β = ∑ i = 1 n x i α i = ( α 1 α 2 ⋯ α n ) [ x 1 x 2 ⋮ x n ]
\beta = \sum^n_{i = 1}x_i \alpha_i=(\alpha_1\ \ \alpha_2\ \ \cdots \ \ \alpha_n)
\left[
\begin{matrix}
x_1 \\
x_2 \\
\vdots \\
x_n
\end{matrix}
\right]
β = i = 1 ∑ n x i α i = ( α 1 α 2 ⋯ α n ) ⎣ ⎢ ⎢ ⎢ ⎡ x 1 x 2 ⋮ x n ⎦ ⎥ ⎥ ⎥ ⎤ x 1 , x 2 , ⋯ , x n x_1,\ x_2,\ \cdots,\ x_n x 1 , x 2 , ⋯ , x n β \beta β { 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 β \beta β 座標 。
若線性空間V n ( F ) V_n(F) V n ( F ) F n F^n F n σ \sigma σ σ \sigma σ σ ( α + β ) = σ ( α ) + σ ( β ) \sigma(\alpha + \beta) = \sigma(\alpha) + \sigma(\beta) σ ( α + β ) = σ ( α ) + σ ( β )
σ ( k α ) = k σ ( α ) \sigma(k \alpha) = k \sigma(\alpha) σ ( k α ) = k σ ( α ) V n ( F ) V_n(F) V n ( F ) F n F^n F n 同構
設{ α 1 , α 2 , ⋯ , α n } \{\alpha_1,\ \alpha_2,\ \cdots,\ \alpha_n\} { α 1 , α 2 , ⋯ , α n } V n ( F ) V_n(F) V n ( F ) V n ( F ) V_n(F) V n ( F ) β i \beta_i β i X i , i = 1 , 2 , ⋯ , m X_i,\ i=1,\ 2,\ \ \cdots,\ m X i , i = 1 , 2 , ⋯ , m V n ( F ) V_n(F) V n ( F ) { β 1 , β 2 , ⋯ , β n } \{\beta_1,\ \beta_2,\ \cdots,\ \beta_n\} { β 1 , β 2 , ⋯ , β n } { X 1 , X 2 , ⋯ , X m } \{X_1,\ X_2,\ \cdots,\ X_m\} { X 1 , X 2 , ⋯ , X m } F n F^n F n
設{ 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 } V n ( F ) V_n(F) V n ( F ) C ∈ F n ∗ n C \in F^{n * n} C ∈ F n ∗ n ( β 1 β 2 ⋯ β n ) = ( α 1 α 2 ⋯ α n ) C (\beta_1 \quad \beta_2 \quad \cdots \quad \beta_n) = (\alpha_1 \quad \alpha_2 \quad \cdots \quad \alpha_n)C ( β 1 β 2 ⋯ β n ) = ( α 1 α 2 ⋯ α n ) C { 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 } 過渡矩陣（基變換矩陣） 。
設向量α ∈ V n ( F ) \alpha \in V_n(F) α ∈ V n ( F ) α \alpha α α = ( α 1 α 2 ⋯ α n ) X \alpha = (\alpha_1 \quad \alpha_2 \quad \cdots \quad \alpha_n)X α = ( α 1 α 2 ⋯ α n ) X
α = ( β 1 β 2 ⋯ β n ) Y \alpha = (\beta_1 \quad \beta_2 \quad \cdots \quad \beta_n)Y α = ( β 1 β 2 ⋯ β n ) Y
因此α = ( β 1 β 2 ⋯ β n ) Y = ( α 1 α 2 ⋯ α n ) C Y \alpha = (\beta_1 \quad \beta_2 \quad \cdots \quad \beta_n)Y = (\alpha_1 \quad \alpha_2 \quad \cdots \quad \alpha_n)CY α = ( β 1 β 2 ⋯ β n ) Y = ( α 1 α 2 ⋯ α n ) C Y X = C Y X=CY X = C Y
例題一 例題二
子空間數學表示：∃ W ⊂ V 且 W ≠ ∅ \exist W \subset V且W \neq \emptyset ∃ W ⊂ V 且 W = ∅ ∀ α , β ∈ W ， 有 α + β ∈ W \forall \alpha,\ \beta \in W，有\alpha+\beta \in W ∀ α , β ∈ W ， 有 α + β ∈ W
∀ k ∈ F ， α ∈ W ， 有 k α ∈ W \forall k \in F，\alpha \in W，有k\alpha \in W ∀ k ∈ F ， α ∈ W ， 有 k α ∈ W 子空間 。任何線性空間都有兩個平凡子空間：一個是它自身V ⊂ V V \subset V V ⊂ V W = { 0 } W=\{0\} W = { 0 }
設V n ( F ) V_n(F) V n ( F ) α 1 , α 2 , ⋯ , α m \alpha_1,\ \alpha_2,\ \cdots,\ \alpha_m α 1 , α 2 , ⋯ , α m L { α 1 , α 2 , ⋯ , α m } = { α ∣ α = ∑ i = 1 m k i α i , k i ∈ F } L\{\alpha_1,\ \alpha_2,\ \cdots,\ \alpha_m\} = \{\alpha | \alpha = \sum^{m}_{i=1}k_i \alpha_i,\ k_i \in F\} L { α 1 , α 2 , ⋯ , α m } = { α ∣ α = i = 1 ∑ m k i α i , k i ∈ F } α 1 , α 2 , ⋯ , α m \alpha_1,\ \alpha_2,\ \cdots,\ \alpha_m α 1 , α 2 , ⋯ , α m L和span等價
直和子空間： W = W 1 ⨁ W 2 W = W_1 \bigoplus W_2 W = W 1 ⨁ W 2 W = W 1 + W 2 W = W_1 + W_2 W = W 1 + W 2 W 1 ∩ W 2 = { 0 } W_1 \cap W_2 = \{0\} W 1 ∩ W 2 = { 0 } W 1 W_1 W 1 W 2 W_2 W 2
直和補子空間： V = W ⨁ U V = W \bigoplus U V = W ⨁ U 直和補子空間 α 1 , α 2 , ⋯ , α r \alpha_1,\ \alpha_2,\ \cdots,\ \alpha_r α 1 , α 2 , ⋯ , α r { α 1 , α 2 , ⋯ , α r , β r + 1 , ⋯ , β n } \{\alpha_1,\ \alpha_2,\ \cdots,\ \alpha_r,\ \beta_{r+1},\ \cdots,\ \beta_n\} { α 1 , α 2 , ⋯ , α r , β r + 1 , ⋯ , β n }
設U = L { β r + 1 , ⋯ , β n } \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad U=L\{\beta_{r+1},\ \cdots,\ \beta_n\} U = L { β r + 1 , ⋯ , β n } V = W ⨁ U V = W \bigoplus U V = W ⨁ U 直和補子空間 。若V n ( F ) 的 基 爲 ： { α 1 , α 2 , ⋯ , α n } V_n(F)的基爲：\{\alpha_1,\ \alpha_2,\ \cdots,\ \alpha_n\} V n ( F ) 的 基 爲 ： { α 1 , α 2 , ⋯ , α n } V = L { α 1 , α 2 , ⋯ , α n } = L { α 1 } ⨁ L { α 2 } ⨁ ⋯ ⨁ L { α n } V=L\{\alpha_1,\ \alpha_2,\ \cdots,\ \alpha_n\}=L\{\alpha_1\} \bigoplus L\{\alpha_2\} \bigoplus \cdots \bigoplus L\{\alpha_n\} V = L { α 1 , α 2 , ⋯ , α n } = L { α 1 } ⨁ L { α 2 } ⨁ ⋯ ⨁ L { α n }
W 1 W_1 W 1 W 2 W_2 W 2 W = W 1 + W 2 W = W_1 + W_2 W = W 1 + W 2
W = W 1 ⨁ W 2 W = W_1 \bigoplus W_2 W = W 1 ⨁ W 2 ∀ X ∈ W ， X 表 示 式 唯 一 ： X = X 1 + X 2 ， 其 中 X 1 ∈ W 1 ， X 2 ∈ W 2 \forall X \in W，X表示式唯一：X = X_1 + X_2，其中X_1 \in W_1，X_2 \in W_2 ∀ X ∈ W ， X 表 示 式 唯 一 ： X = X 1 + X 2 ， 其 中 X 1 ∈ W 1 ， X 2 ∈ W 2 W中零向量表達式唯一，即只要0 = X 1 + X 2 ， X 1 ∈ W 1 ， X 2 ∈ W 2 0 = X_1 + X_2，X_1 \in W_1，X_2 \in W_2 0 = X 1 + X 2 ， X 1 ∈ W 1 ， X 2 ∈ W 2 X 1 = 0 ， X 2 = 0 X_1 = 0，X_2 = 0 X 1 = 0 ， X 2 = 0
d i m W = d i m W 1 + d i m W 2 dimW = dimW_1 + dimW_2 d i m W = d i m W 1 + d i m W 2
正交子空間： U ⊥ = { α ∣ α ∈ V , ∀ β ∈ U , ( α , β ) = 0 } U^\perp = \{\alpha | \alpha \in V,\ \forall \beta \in U,\ (\alpha,\ \beta) = 0\} U ⊥ = { α ∣ α ∈ V , ∀ β ∈ U , ( α , β ) = 0 } 正交補子空間： V n = U + U ⊥ V_n = U + U^ \perp V n = U + U ⊥ U ⊥ U^ \perp U ⊥ U U U
線性變換T，W是子空間，若∀ α ∈ W , 有 T ( α ) ∈ W \forall \alpha \in W,\ 有T(\alpha) \in W ∀ α ∈ W , 有 T ( α ) ∈ W T ( W ) ⊂ W T(W) \subset W T ( W ) ⊂ W
零空間N(A)： N ( A ) = { X ∣ A X = 0 } ⊂ F n N(A) = \{X | AX = 0\} \subset F^n \quad N ( A ) = { X ∣ A X = 0 } ⊂ F n 等價於求方程組的非零解 列空間R(A)： R ( A ) = L { α 1 , α 2 , ⋯ , α n } = { y = A x ∣ x = ( x 1 , x 2 , ⋯ , x n ) T ∈ F n } = { x 1 α 1 + x 2 α 2 + ⋯ + x n α n ∣ x i ∈ F } ⊂ F m R(A) = L\{\alpha_1,\ \alpha_2,\ \cdots,\ \alpha_n\} = \{y=Ax | x = (x_1,\ x_2,\ \cdots,\ x_n) ^ T \in F^n\} = \{x_1 \alpha_1 + x_2 \alpha_2 + \cdots + x_n \alpha_n | x_i \in F\} \subset F^m \quad R ( A ) = L { α 1 , α 2 , ⋯ , α n } = { y = A x ∣ x = ( x 1 , x 2 , ⋯ , x n ) T ∈ F n } = { x 1 α 1 + x 2 α 2 + ⋯ + x n α n ∣ x i ∈ F } ⊂ F m 等價於求方程組的極大線性無關組 當T是線性空間V n ( F ) V_n(F) V n ( F ) 像空間： R ( T ) = { β ∣ ∃ α ∈ V n ( F ) , β = T ( α ) } R(T) = \{\beta | \exist \alpha \in V_n(F),\ \beta = T(\alpha) \} R ( T ) = { β ∣ ∃ α ∈ V n ( F ) , β = T ( α ) } V n ( F ) V_n(F) V n ( F ) 像空間 \quad 等價於求T ( α ) T(\alpha) T ( α ) 零空間： N ( T ) = { α ∣ T ( α ) = 0 } N(T) = \{\alpha | T(\alpha) = 0 \} N ( T ) = { α ∣ T ( α ) = 0 } V n ( F ) V_n(F) V n ( F ) 零空間 。\quad 等價於求T ( α ) = 0 時 α 的 解 T(\alpha) = 0時\alpha的解 T ( α ) = 0 時 α 的 解 { α 1 , α 2 , ⋯ , α n } \{\alpha_1,\ \alpha_2,\ \cdots,\ \alpha_n\} { α 1 , α 2 , ⋯ , α n } R ( T ) = L { T ( α 1 ) , T ( α 2 ) , ⋯ , T ( α n ) } R(T) = L\{T(\alpha_1),\ T(\alpha_2),\ \cdots,\ T(\alpha_n)\} R ( T ) = L { T ( α 1 ) , T ( α 2 ) , ⋯ , T ( α n ) }
交空間： W 1 ∩ W 2 = { α ∣ α ∈ W 1 且 α ∈ W 2 } W_1 \cap W_2 = \{\alpha | \alpha \in W_1且\alpha \in W_2\} W 1 ∩ W 2 = { α ∣ α ∈ W 1 且 α ∈ W 2 } 和空間： W 1 + W 2 = { α ∣ α = α 1 + α 2 , α 1 ∈ W 1 , α 2 ∈ W 2 } W_1 + W_2 = \{\alpha | \alpha = \alpha_1 + \alpha_2,\ \alpha_1 \in W_1,\ \alpha_2 \in W_2\} W 1 + W 2 = { α ∣ α = α 1 + α 2 , α 1 ∈ W 1 , α 2 ∈ W 2 }
d i m W 1 + d i m W 2 = d i m ( W 1 + W 2 ) + d i m ( W 1 ∩ W 2 ) dimW_1 + dimW_2 = dim(W_1 + W_2) + dim(W_1 \cap W_2) d i m W 1 + d i m W 2 = d i m ( W 1 + W 2 ) + d i m ( W 1 ∩ W 2 ) W 1 W_1 W 1 W 2 W_2 W 2
V n ( F ) → F : V n ( F ) → F V_n(F) \rightarrow F: V_n(F) \rightarrow F V n ( F ) → F : V n ( F ) → F ( α , β ) = ( β , α ‾ ) (\alpha,\ \beta) = (\overline{\beta,\ \alpha}) ( α , β ) = ( β , α ) ( β , α ‾ ) (\overline{\beta,\ \alpha}) ( β , α ) ( β , α ) (\beta,\ \alpha) ( β , α ) ( k α , β ) = k ( α , β ) (k \alpha,\ \beta) = k(\alpha,\ \beta) ( k α , β ) = k ( α , β ) ( α 1 + α 2 , β ) = ( α 1 , β ) + ( α 2 , β ) \qquad \qquad (\alpha_1 + \alpha_2,\ \beta) = (\alpha_1,\ \beta) + (\alpha_2,\ \beta) ( α 1 + α 2 , β ) = ( α 1 , β ) + ( α 2 , β ) ( α , α ) ≥ 0 , ( α , α ) = 0 (\alpha,\ \alpha) \geq 0,\ (\alpha,\ \alpha)=0 ( α , α ) ≥ 0 , ( α , α ) = 0 α = 0 \alpha = 0 α = 0 ( α , β ) (\alpha,\ \beta) ( α , β ) V n ( F ) V_n(F) V n ( F ) 內積 ，[ V n ( F ) ; ( α , β ) ] [V_n(F);\ (\alpha,\ \beta)] [ V n ( F ) ; ( α , β ) ] 內積空間
歐式空間： 實數域R上的內積空間酉空間： 複數域C上的內積空間共軛： 實部相同，虛部取相反數轉置共軛矩陣： A H = ( A ‾ ) T A^H = (\overline{A})^T A H = ( A ) T 向量夾角： θ = a r c c o s ( α , β ) ∣ ∣ α ∣ ∣ ∣ ∣ β ∣ ∣ \theta = arccos \frac{(\alpha,\ \beta)}{||\alpha|| \ ||\beta||} θ = a r c c o s ∣ ∣ α ∣ ∣ ∣ ∣ β ∣ ∣ ( α , β ) 向量正交： ( α , β ) = 0 (\alpha,\ \beta) = 0 ( α , β ) = 0 標準正交向量組： 向量組{ α 1 , α 2 , ⋯ , α m } \{\alpha_1,\ \alpha_2,\ \cdots,\ \alpha_m\} { α 1 , α 2 , ⋯ , α m } ( α i , α j ) = { 1 i = j 0 i ≠ j (\alpha_i,\ \alpha_j) =
\begin{cases}
1 & i = j \\
0 & i \neq j
\end{cases} ( α i , α j ) = { 1 0 i = j i = j Gram-Schmidt正交化(求正交向量組)： β k = α k − ∑ i = 1 k − 1 ( α k , β i ) ( β i , β i ) β i , k = 1 , 2 , ⋯ , m \beta_k = \alpha_k - \sum^{k - 1}_{i = 1} \frac{(\alpha_k,\ \beta_i)}{(\beta_i,\ \beta_i)} \beta_i,\quad k = 1,\ 2,\ \cdots,\ m β k = α k − i = 1 ∑ k − 1 ( β i , β i ) ( α k , β i ) β i , k = 1 , 2 , ⋯ , m
其中由α i {\alpha_i} α i β k \beta_k β k
向量的長度 也叫做向量的歐幾里得範數 ，即∣ ∣ α ∣ ∣ = ( α , α ) ||\alpha|| = \sqrt{(\alpha,\ \alpha)} ∣ ∣ α ∣ ∣ = ( α , α ) 單位向量： ∣ ∣ α ∣ ∣ = 1 ||\alpha|| = 1 ∣ ∣ α ∣ ∣ = 1
∣ ∣ k α ∣ ∣ = ∣ ∣ k ∣ ∣ ⋅ ∣ ∣ α ∣ ∣ ||k \alpha|| = ||k|| \cdot ||\alpha|| ∣ ∣ k α ∣ ∣ = ∣ ∣ k ∣ ∣ ⋅ ∣ ∣ α ∣ ∣
∣ ∣ α + β ∣ ∣ ≤ ∣ ∣ α ∣ ∣ + ∣ ∣ β ∣ ∣ ||\alpha + \beta|| \leq ||\alpha|| + ||\beta|| ∣ ∣ α + β ∣ ∣ ≤ ∣ ∣ α ∣ ∣ + ∣ ∣ β ∣ ∣
∀ α ≠ 0 , α 0 = α ∣ ∣ α ∣ ∣ , ∣ ∣ α 0 ∣ ∣ = 1 , 取 α 0 = α ∣ ∣ α ∣ ∣ 的 過 程 稱 爲 標 準 化 \forall \alpha \neq 0,\ \alpha^0 = \frac{\alpha}{||\alpha||},\ ||\alpha^0|| = 1,\ 取\alpha^0 = \frac{\alpha}{||\alpha||}的過程稱爲標準化 ∀ α = 0 , α 0 = ∣ ∣ α ∣ ∣ α , ∣ ∣ α 0 ∣ ∣ = 1 , 取 α 0 = ∣ ∣ α ∣ ∣ α 的 過 程 稱 爲 標 準 化
α 和 β 線 性 相 關 \alpha和\beta線性相關 α 和 β 線 性 相 關
⟺ \iff ⟺
[ V n ( F ) ; ( α , β ) ] 爲 內 積 空 間 ， ∀ α , β ∈ V n ( F ) ， 有 ∣ ( α , β ) ∣ 2 ≤ ( α , α ) ( β , β ) [V_n(F);\ (\alpha,\ \beta)]爲內積空間，\forall \alpha, \ \beta \in V_n(F)，有|(\alpha,\ \beta)|^2 \leq (\alpha,\ \alpha) (\beta,\ \beta) [ V n ( F ) ; ( α , β ) ] 爲 內 積 空 間 ， ∀ α , β ∈ V n ( F ) ， 有 ∣ ( α , β ) ∣ 2 ≤ ( α , α ) ( β , β )
C n , ( α , β ) = β H α ⇒ ∣ ∑ i = 1 N x i y i ‾ ∣ 2 ≤ ∑ i = 1 n ∣ x i ∣ 2 ⋅ ∑ i = 1 n ∣ y i ∣ 2 C^n,\ (\alpha,\ \beta) = \beta^H \alpha \Rightarrow |\sum^{N}_{i = 1} x_i \overline{y_i}|^2 \leq \sum^{n}_{i=1} |x_i|^2 \cdot \sum^{n}_{i=1} |y_i|^2 C n , ( α , β ) = β H α ⇒ ∣ i = 1 ∑ N x i y i ∣ 2 ≤ i = 1 ∑ n ∣ x i ∣ 2 ⋅ i = 1 ∑ n ∣ y i ∣ 2
C n ∗ n , ( A , B ) = t r ( B H A ) ⇒ ∣ t r ( B H A ) 2 ∣ ≤ t r ( A H A ) ⋅ t r ( B H B ) C^{n * n},\ (A,\ B) = tr(B^HA) \Rightarrow |tr(B^H A)^2| \leq tr(A^H A) \cdot tr(B^H B) C n ∗ n , ( A , B ) = t r ( B H A ) ⇒ ∣ t r ( B H A ) 2 ∣ ≤ t r ( A H A ) ⋅ t r ( B H B )
柯西不等式可寫爲∣ α , β ∣ ≤ ∣ ∣ α ∣ ∣ ⋅ ∣ ∣ β ∣ ∣ |\alpha,\ \beta| \leq ||\alpha|| \cdot ||\beta|| ∣ α , β ∣ ≤ ∣ ∣ α ∣ ∣ ⋅ ∣ ∣ β ∣ ∣
變換： 線性空間V n ( F ) V_n(F) V n ( F ) ∀ α ∈ V n ( F ) \forall \alpha \in V_n(F) ∀ α ∈ V n ( F ) α ′ = T ( α ) ∈ V n ( F ) \alpha' = T(\alpha) \in V_n(F) α ′ = T ( α ) ∈ V n ( F ) 線性變換： 變換T同時滿足∀ α , β ∈ V n ( F ) , T ( α + β ) = T ( α ) + T ( β ) \forall \alpha,\ \beta \in V_n(F), \quad T(\alpha + \beta) = T(\alpha) + T(\beta) ∀ α , β ∈ V n ( F ) , T ( α + β ) = T ( α ) + T ( β )
∀ k ∈ F , ∀ α ∈ V n ( F ) , T ( k α ) = k T ( α ) \forall k \in F,\ \forall \alpha \in V_n(F), \quad T(k \alpha) = k T(\alpha) ∀ k ∈ F , ∀ α ∈ V n ( F ) , T ( k α ) = k T ( α )
可將上述兩式合寫爲T ( k 1 α 1 + k 2 α 2 ) = k 1 T ( α 1 ) + k 2 T ( α 2 ) T(k_1 \alpha_1 + k_2 \alpha_2) = k_1 T(\alpha_1) + k_2 T(\alpha_2) T ( k 1 α 1 + k 2 α 2 ) = k 1 T ( α 1 ) + k 2 T ( α 2 ) 線性變換與同構：
零變換： ∀ α ∈ V n ( F ) , T ( α ) = 0 \forall \alpha \in V_n(F),\ T(\alpha) = 0 ∀ α ∈ V n ( F ) , T ( α ) = 0 恆等變換： ∀ α ∈ V n ( F ) , T ( α ) = α \forall \alpha \in V_n(F),\ T(\alpha) = \alpha ∀ α ∈ V n ( F ) , T ( α ) = α 矩陣： T是V n ( F ) V_n(F) V n ( F ) { α 1 , α 2 , ⋯ , α n } \{\alpha_1,\ \alpha_2,\ \cdots, \alpha_n\} { α 1 , α 2 , ⋯ , α n } ∃ A ∈ F n ∗ n \exist A \in F^{n * n} ∃ A ∈ F n ∗ n T ( α 1 , α 2 , ⋯ , α n ) = ( α 1 , α 2 , ⋯ , α n ) A T(\alpha_1,\ \alpha_2,\ \cdots, \alpha_n) = (\alpha_1,\ \alpha_2,\ \cdots, \alpha_n) A T ( α 1 , α 2 , ⋯ , α n ) = ( α 1 , α 2 , ⋯ , α n ) A { α 1 , α 2 , ⋯ , α n } \{\alpha_1,\ \alpha_2,\ \cdots, \alpha_n\} { α 1 , α 2 , ⋯ , α n }
T ( 0 ) = 0 T(0) = 0 T ( 0 ) = 0
T ( − α ) = − T ( α ) T(- \alpha) = - T(\alpha) T ( − α ) = − T ( α )
T ∑ i = 1 r k i α i = ∑ i = 1 r k i T ( α i ) T \sum^r_{i = 1} k_i \alpha_i = \sum^r_{i = 1} k_i T(\alpha_i) T ∑ i = 1 r k i α i = ∑ i = 1 r k i T ( α i )
若{ α 1 , α 2 , ⋯ , α s } \{\alpha_1,\quad \alpha_2,\quad \cdots,\quad \alpha_s \} { α 1 , α 2 , ⋯ , α s } { T ( α 1 ) , T ( α 2 ) , ⋯ , T ( α s ) } \{T(\alpha_1),\quad T(\alpha_2),\quad \cdots,\quad T(\alpha_s) \} { T ( α 1 ) , T ( α 2 ) , ⋯ , T ( α s ) }
T 1 和 T 2 T_1和T_2 T 1 和 T 2 { α 1 , α 2 , ⋯ , α n } \{\alpha_1,\ \alpha_2,\ \cdots, \alpha_n\} { α 1 , α 2 , ⋯ , α n } A 1 和 A 2 A_1和A_2 A 1 和 A 2 { α 1 , α 2 , ⋯ , α n } \{\alpha_1,\ \alpha_2,\ \cdots, \alpha_n\} { α 1 , α 2 , ⋯ , α n }
T 1 + T 2 T_1 + T_2 T 1 + T 2 ( A 1 + A 2 ) (A_1 + A_2) ( A 1 + A 2 )
T 1 T 2 T_1 T_2 T 1 T 2 A 1 A 2 A_1 A_2 A 1 A 2
k T 1 k T_1 k T 1 k A 1 k A_1 k A 1
T 1 T_1 T 1 ⟺ \iff ⟺ A 1 A_1 A 1 T 1 − 1 T^{-1}_1 T 1 − 1 A 1 − 1 A^{-1}_1 A 1 − 1
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 } B = C − 1 A C B = C^{-1} A C B = C − 1 A C
例題一 例題二
內積空間上的正交變換： ∀ α , β ∈ [ V n ( F ) ; ( α , β ) ] \forall \alpha,\ \beta \in [V_n(F);\ (\alpha,\ \beta)] ∀ α , β ∈ [ V n ( F ) ; ( α , β ) ] ( T ( α ) , T ( β ) ) = ( α , β ) (T(\alpha),\ T(\beta)) = (\alpha,\ \beta) ( T ( α ) , T ( β ) ) = ( α , β ) 正交變換： 當內積空間爲歐式空間 時，變換T爲正交變換酉變換： 當內積空間爲酉空間 時，變換T爲酉變換
T是內積空間上的線性變換，則下列命題等價：
T是正交(酉)變換
T保持向量長度不變
T把空間V n ( F ) V_n(F) V n ( F )
正交變換關於任一標準正交基的矩陣C滿足C T C = C C T = I C^T C = C C^T = I C T C = C C T = I U H U = U U H = I U^H U = U U^H = I U H U = U U H = I
正交矩陣( C ) (C) ( C ) ± 1 \pm1 ± 1 ( U ) (U) ( U ) C − 1 = C T ; U − 1 = U H C^{-1} = C^T;\ U^{-1} = U^H C − 1 = C T ; U − 1 = U H R n ( C n ) R^n(C^n) R n ( C n )
例題一 本博客中所有內容均來自《矩陣論》第二版 華科，加入了一些自己的理解和簡化，如有侵權，請聯繫刪除
