线性代数
行列式
1.行列式按行(列)展开定理
(1)
设 A = ( a ij ) n × n A = \left( a_{\text{ij}} \right)_{n \times n} A=(aij)n×n,则:$a_{i1}A_{j1} + a_{i2}A_{j2} + \cdots + a_{\text{in}}A_{\text{jn}} = \left{ \begin{matrix}
& \left| A \right|,i = j \
& 0,i \neq j \
\end{matrix} \right.\ $
或$a_{1i}A_{1j} + a_{2i}A_{2j} + \cdots + a_{\text{ni}}A_{\text{nj}} = \left{ \begin{matrix}
& \left| A \right|,i = j \
& 0,i \neq j \
\end{matrix} \right.\ $
即
A A ∗ = A ∗ A = ∣ A ∣ E , AA^{\ast} = A^{\ast}A = \left| A \right|E, AA∗=A∗A=∣A∣E,其中: A ∗ = ( A 11 A 12 … A 1 n A 21 A 22 … A 2 n … … … … A n 1 A n 2 … A nn ) = ( A ji ) = ( A ij ) T A^{\ast} = \begin{pmatrix} A_{11} & A_{12} & \ldots & A_{1n} \\ A_{21} & A_{22} & \ldots & A_{2n} \\ \ldots & \ldots & \ldots & \ldots \\ A_{n1} & A_{n2} & \ldots & A_{\text{nn}} \\ \end{pmatrix} = (A_{\text{ji}}) = {(A_{\text{ij}})}^{T} A∗=
A11A21…An1A12A22…An2…………A1nA2n…Ann
=(Aji)=(Aij)T
D n = ∣ 1 1 … 1 x 1 x 2 … x n … … … … x 1 n − 1 x 2 n − 1 … x n n − 1 ∣ = = ∏ 1 ≤ j < i ≤ n ( x i − x j ) D_{n} = \begin{vmatrix} 1 & 1 & \ldots & 1 \\ x_{1} & x_{2} & \ldots & x_{n} \\ \ldots & \ldots & \ldots & \ldots \\ x_{1}^{n - 1} & x_{2}^{n - 1} & \ldots & x_{n}^{n - 1} \\ \end{vmatrix} = = \prod_{1 \leq j < i \leq n}^{}\,(x_{i} - x_{j}) Dn= 1x1…x1n−11x2…x2n−1…………1xn…xnn−1 ==∏1≤j<i≤n(xi−xj)
(2)
设 A , B A,B A,B为 n n n阶方阵,则 ∣ AB ∣ = ∣ A ∣ ∣ B ∣ = ∣ B ∣ ∣ A ∣ = ∣ BA ∣ \left| \text{AB} \right| = \left| A \right|\left| B \right| = \left| B \right|\left| A \right| = \left| \text{BA} \right| ∣AB∣=∣A∣∣B∣=∣B∣∣A∣=∣BA∣,但 ∣ A ± B ∣ = ∣ A ∣ ± ∣ B ∣ \left| A \pm B \right| = \left| A \right| \pm \left| B \right| ∣A±B∣=∣A∣±∣B∣不一定成立。
(3) ∣ kA ∣ = k n ∣ A ∣ \left| \text{kA} \right| = k^{n}\left| A \right| ∣kA∣=kn∣A∣, A A A为 n n n阶方阵。
(4)
设 A A A为 n n n阶方阵, ∣ A T ∣ = ∣ A ∣ ; ∣ A − 1 ∣ = ∣ A ∣ − 1 |A^{T}| = |A|;|A^{- 1}| = |A|^{- 1} ∣AT∣=∣A∣;∣A−1∣=∣A∣−1(若 A A A可逆), ∣ A ∗ ∣ = ∣ A ∣ n − 1 |A^{\ast}| = |A|^{n - 1} ∣A∗∣=∣A∣n−1
n ≥ 2 n \geq 2 n≥2
(5) $\left| \begin{matrix}
& \text{A\quad O} \
& \text{O\quad B} \
\end{matrix} \right| = \left| \begin{matrix}
& \text{A\quad C} \
& \text{O\quad B} \
\end{matrix} \right| = \left| \begin{matrix}
& \text{A\quad O} \
& \text{C\quad B} \
\end{matrix} \right| = \left| A||B| \right.\ $
, A , B A,B A,B为方阵,但 ∣ O A m × m B n × n O ∣ = ( − 1 ) mn ∣ A ∣ ∣ B ∣ \left| \begin{matrix} & \text{O\quad\quad}A_{m \times m} \\ & B_{n \times n}\text{\quad O} \\ \end{matrix} \right| = ({- 1)}^{\text{mn}}|A||B|
OAm×mBn×nO
=(−1)mn∣A∣∣B∣ 。
(6) 范德蒙行列式 D n = ∣ 1 1 … 1 x 1 x 2 … x n … … … … x 1 n − 1 x 2 n − 1 … x n n − 1 ∣ = = ∏ 1 ≤ j < i ≤ n ( x i − x j ) D_{n} = \begin{vmatrix} 1 & 1 & \ldots & 1 \\ x_{1} & x_{2} & \ldots & x_{n} \\ \ldots & \ldots & \ldots & \ldots \\ x_{1}^{n - 1} & x_{2}^{n - 1} & \ldots & x_{n}^{n - 1} \\ \end{vmatrix} = = \prod_{1 \leq j < i \leq n}^{}\,(x_{i} - x_{j}) Dn= 1x1…x1n−11x2…x2n−1…………1xn…xnn−1 ==∏1≤j<i≤n(xi−xj)
设 A A A是 n n n阶方阵, λ i ( i = 1 , 2 ⋯ , n ) \lambda_{i}(i = 1,2\cdots,n) λi(i=1,2⋯,n)是 A A A的 n n n个特征值,则
∣ A ∣ = ∏ i = 1 n λ i |A| = \prod_{i = 1}^{n}\lambda_{i} ∣A∣=∏i=1nλi
矩阵
矩阵: m × n m \times n m×n个数 a ij a_{\text{ij}} aij排成 m m m行 n n n列的表格 [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋯ ⋯ ⋯ ⋯ ⋯ a m 1 a m 2 ⋯ a mn ] \begin{bmatrix} & a_{11}\quad a_{12}\quad\cdots\quad a_{1n} \\ & a_{21}\quad a_{22}\quad\cdots\quad a_{2n} \\ & \quad\cdots\cdots\cdots\cdots\cdots \\ & a_{m1}\quad a_{m2}\quad\cdots\quad a_{\text{mn}} \\ \end{bmatrix}
a11a12⋯a1na21a22⋯a2n⋯⋯⋯⋯⋯am1am2⋯amn
称为矩阵,简记为 A A A,或者 ( a ij ) m × n \left( a_{\text{ij}} \right)_{m \times n} (aij)m×n
。若 m = n m = n m=n,则称 A A A是 n n n阶矩阵或 n n n阶方阵。
**矩阵的线性运算 **
**1.矩阵的加法 **
设 A = ( a ij ) , B = ( b ij ) A = (a_{\text{ij}}),B = (b_{\text{ij}}) A=(aij),B=(bij)是两个 m × n m \times n m×n矩阵,则 m × n m \times n m×n
矩阵 C = ( c ij ) = a ij + b ij C = (c_{\text{ij}}) = a_{\text{ij}} + b_{\text{ij}} C=(cij)=aij+bij称为矩阵 A A A与 B B B的和,记为 A + B = C A + B = C A+B=C
。
**2.矩阵的数乘 **
设 A = ( a ij ) A = (a_{\text{ij}}) A=(aij)是 m × n m \times n m×n矩阵, k k k是一个常数,则 m × n m \times n m×n矩阵 ( k a ij ) (ka_{\text{ij}}) (kaij)称为数 k k k与矩阵 A A A的数乘,记为 kA \text{kA} kA。
**3.矩阵的乘法 **
设 A = ( a ij ) A = (a_{\text{ij}}) A=(aij)是 m × n m \times n m×n矩阵, B = ( b ij ) B = (b_{\text{ij}}) B=(bij)是 n × s n \times s n×s矩阵,那么 m × s m \times s m×s矩阵 C = ( c ij ) C = (c_{\text{ij}}) C=(cij),其中
c ij = a i 1 b 1 j + a i 2 b 2 j + ⋯ + a in b nj = ∑ k = 1 n a ik b kj c_{\text{ij}} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{\text{in}}b_{\text{nj}} = \sum_{k = 1}^{n}{a_{\text{ik}}b_{\text{kj}}} cij=ai1b1j+ai2b2j+⋯+ainbnj=∑k=1naikbkj称为 AB \text{AB} AB的乘积,记为 C = A B C = AB C=AB
。
4.
A T \mathbf{A}^{\mathbf{T}} AT、 A − 1 \mathbf{A}^{\mathbf{- 1}} A−1、 A ∗ \mathbf{A}^{\mathbf{\ast}} A∗**三者之间的关系
**
(1)
( A T ) T = A , ( A B ) T = B T A T , ( k A ) T = k A T , ( A ± B ) T = A T ± B T {(A^{T})}^{T} = A,{(AB)}^{T} = B^{T}A^{T},{(kA)}^{T} = kA^{T},{(A \pm B)}^{T} = A^{T} \pm B^{T} (AT)T=A,(AB)T=BTAT,(kA)T=kAT,(A±B)T=AT±BT
(2)
( A − 1 ) − 1 = A , ( AB ) − 1 = B − 1 A − 1 , ( kA ) − 1 = 1 k A − 1 , \left( A^{- 1} \right)^{- 1} = A,\left( \text{AB} \right)^{- 1} = B^{- 1}A^{- 1},\left( \text{kA} \right)^{- 1} = \frac{1}{k}A^{- 1}, (A−1)−1=A,(AB)−1=B−1A−1,(kA)−1=k1A−1,
但 ( A ± B ) − 1 = A − 1 ± B − 1 {(A \pm B)}^{- 1} = A^{- 1} \pm B^{- 1} (A±B)−1=A−1±B−1不一定成立。
(3)
( A ∗ ) ∗ = ∣ A ∣ n − 2 A ( n ≥ 3 ) \left( A^{\ast} \right)^{\ast} = |A|^{n - 2}\ A\ \ (n \geq 3) (A∗)∗=∣A∣n−2 A (n≥3), ( AB ) ∗ = B ∗ A ∗ , \left( \text{AB} \right)^{\ast} = B^{\ast}A^{\ast}, (AB)∗=B∗A∗,
( kA ) ∗ = k n − 1 A ∗ ( n ≥ 2 ) \left( \text{kA} \right)^{\ast} = k^{n - 1}A^{\ast}\text{\ \ }\left( n \geq 2 \right) (kA)∗=kn−1A∗ (n≥2)
但 ( A ± B ) ∗ = A ∗ ± B ∗ \left( A \pm B \right)^{\ast} = A^{\ast} \pm B^{\ast} (A±B)∗=A∗±B∗不一定成立。
(4)
( A − 1 ) T = ( A T ) − 1 , ( A − 1 ) ∗ = ( A A ∗ ) − 1 , ( A ∗ ) T = ( A T ) ∗ {(A^{- 1})}^{T} = {(A^{T})}^{- 1},\ \left( A^{- 1} \right)^{\ast} = {(AA^{\ast})}^{- 1},{(A^{\ast})}^{T} = \left( A^{T} \right)^{\ast} (A−1)T=(AT)−1, (A−1)∗=(AA∗)−1,(A∗)T=(AT)∗
5.有关 A ∗ \mathbf{A}^{\mathbf{\ast}} A∗**的结论 **
(1) A A ∗ = A ∗ A = ∣ A ∣ E AA^{\ast} = A^{\ast}A = |A|E AA∗=A∗A=∣A∣E
(2)
∣ A ∗ ∣ = ∣ A ∣ n − 1 ( n ≥ 2 ) , ( k A ) ∗ = k n − 1 A ∗ , ( A ∗ ) ∗ = ∣ A ∣ n − 2 A ( n ≥ 3 ) |A^{\ast}| = |A|^{n - 1}\ (n \geq 2),\ \ \ \ {(kA)}^{\ast} = k^{n - 1}A^{\ast},{\text{\ \ }\left( A^{\ast} \right)}^{\ast} = |A|^{n - 2}A(n \geq 3) ∣A∗∣=∣A∣n−1 (n≥2), (kA)∗=kn−1A∗, (A∗)∗=∣A∣n−2A(n≥3)
(3)
若 A A A可逆,则 A ∗ = ∣ A ∣ A − 1 , ( A ∗ ) ∗ = 1 ∣ A ∣ A A^{\ast} = |A|A^{- 1},{(A^{\ast})}^{\ast} = \frac{1}{|A|}A A∗=∣A∣A−1,(A∗)∗=∣A∣1A
(4) 若 A A A为 n n n阶方阵,则:
$r(A^{\ast}) = \left{ \begin{matrix}
& n,\quad r(A) = n \
& 1,\quad r(A) = n - 1 \
& 0,\quad r(A) < n - 1 \
\end{matrix} \right.\ $
6.有关 A − 1 \mathbf{A}^{\mathbf{- 1}} A−1的结论
A A A可逆 ⇔ A B = E ; ⇔ ∣ A ∣ ≠ 0 ; ⇔ r ( A ) = n ; \Leftrightarrow AB = E; \Leftrightarrow |A| \neq 0; \Leftrightarrow r(A) = n; ⇔AB=E;⇔∣A∣=0;⇔r(A)=n;
⇔ A \Leftrightarrow A ⇔A可以表示为初等矩阵的乘积;$\Leftrightarrow A; \Leftrightarrow Ax = 0\ $。
7.有关矩阵秩的结论
(1) 秩 r ( A ) r(A) r(A)=行秩=列秩;
(2) r ( A m × n ) ≤ min ( m , n ) ; r(A_{m \times n}) \leq \min(m,n); r(Am×n)≤min(m,n);
(3) A ≠ 0 ⇒ r ( A ) ≥ 1 A \neq 0 \Rightarrow r(A) \geq 1 A=0⇒r(A)≥1;
(4) r ( A ± B ) ≤ r ( A ) + r ( B ) ; r(A \pm B) \leq r(A) + r(B); r(A±B)≤r(A)+r(B);
(5) 初等变换不改变矩阵的秩
(6) r ( A ) + r ( B ) − n ≤ r ( A B ) ≤ min ( r ( A ) , r ( B ) ) , r(A) + r(B) - n \leq r(AB) \leq \min(r(A),r(B)), r(A)+r(B)−n≤r(AB)≤min(r(A),r(B)),特别若 A B = O AB = O AB=O
则: r ( A ) + r ( B ) ≤ n r(A) + r(B) \leq n r(A)+r(B)≤n
(7) 若 A − 1 A^{- 1} A−1存在 ⇒ r ( A B ) = r ( B ) ; \Rightarrow r(AB) = r(B); ⇒r(AB)=r(B); 若 B − 1 B^{- 1} B−1存在
⇒ r ( A B ) = r ( A ) ; \Rightarrow r(AB) = r(A); ⇒r(AB)=r(A);
若 r ( A m × n ) = n ⇒ r ( A B ) = r ( B ) ; r(A_{m \times n}) = n \Rightarrow r(AB) = r(B); r(Am×n)=n⇒r(AB)=r(B);
若 r ( A m × s ) = n ⇒ r ( A B ) = r ( A ) r(A_{m \times s}) = n \Rightarrow r(AB) = r\left( A \right) r(Am×s)=n⇒r(AB)=r(A)。
(8) r ( A m × s ) = n ⇔ A x = 0 r(A_{m \times s}) = n \Leftrightarrow Ax = 0 r(Am×s)=n⇔Ax=0只有零解
8.分块求逆公式
( A O O B ) − 1 = ( A − 1 O O B − 1 ) \begin{pmatrix} A & O \\ O & B \\ \end{pmatrix}^{- 1} = \begin{pmatrix} A^{- 1} & O \\ O & B^{- 1} \\ \end{pmatrix} (AOOB)−1=(A−1OOB−1); ( A C O B ) − 1 = ( A − 1 − A − 1 C B − 1 O B − 1 ) \begin{pmatrix} A & C \\ O & B \\ \end{pmatrix}^{- 1} = \begin{pmatrix} & A^{- 1}\quad - A^{- 1}CB^{- 1} \\ & \text{O\quad\quad\quad}B^{- 1} \\ \end{pmatrix} (AOCB)−1=(A−1−A−1CB−1OB−1);
( A O C B ) − 1 = ( A − 1 O − B − 1 C A − 1 B − 1 ) \begin{pmatrix} A & O \\ C & B \\ \end{pmatrix}^{- 1} = \begin{pmatrix} & A^{- 1}\text{\quad\quad\:\:\quad O} \\ & - B^{- 1}CA^{- 1}\quad B^{- 1} \\ \end{pmatrix} (ACOB)−1=(A−1O−B−1CA−1B−1); ( O A B O ) − 1 = ( O B − 1 A − 1 O ) \begin{pmatrix} O & A \\ B & O \\ \end{pmatrix}^{- 1} = \begin{pmatrix} O & B^{- 1} \\ A^{- 1} & O \\ \end{pmatrix} (OBAO)−1=(OA−1B−1O)
这里 A A A, B B B均为可逆方阵。
向量
**1.有关向量组的线性表示 **
(1)
α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs线性相关 ⇔ \Leftrightarrow ⇔至少有一个向量可以用其余向量线性表示。
(2)
α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs线性无关, α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs, β \beta β线性相关 ⇔ β \Leftrightarrow \beta ⇔β可以由 α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs唯一线性表示。
(3) β \beta β可以由 α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs线性表示
⇔ r ( α 1 , α 2 , ⋯ , α s ) = r ( α 1 , α 2 , ⋯ , α s , β ) \Leftrightarrow r(\alpha_{1},\alpha_{2},\cdots,\alpha_{s}) = r(\alpha_{1},\alpha_{2},\cdots,\alpha_{s},\beta) ⇔r(α1,α2,⋯,αs)=r(α1,α2,⋯,αs,β)
。
**2.有关向量组的线性相关性 **
(1)部分相关,整体相关;整体无关,部分无关.
(2) ① n n n个 n n n维向量
α 1 , α 2 ⋯ α n \alpha_{1},\alpha_{2}\cdots\alpha_{n} α1,α2⋯αn线性无关 ⇔ ∣ [ α 1 α 2 ⋯ α n ] ∣ ≠ 0 \Leftrightarrow \left| \left\lbrack \alpha_{1}\alpha_{2}\cdots\alpha_{n} \right\rbrack \right| \neq 0 ⇔∣[α1α2⋯αn]∣=0,
n n n个 n n n维向量 α 1 , α 2 ⋯ α n \alpha_{1},\alpha_{2}\cdots\alpha_{n} α1,α2⋯αn线性相关
⇔ ∣ [ α 1 , α 2 , ⋯ , α n ] ∣ = 0 \Leftrightarrow |\lbrack\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\rbrack| = 0 ⇔∣[α1,α2,⋯,αn]∣=0
。
② n + 1 n + 1 n+1个 n n n维向量线性相关。
③
若 α 1 , α 2 ⋯ α S \alpha_{1},\alpha_{2}\cdots\alpha_{S} α1,α2⋯αS线性无关,则添加分量后仍线性无关;或一组向量线性相关,去掉某些分量后仍线性相关。
3.有关向量组的线性表示
(1)
α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs线性相关 ⇔ \Leftrightarrow ⇔至少有一个向量可以用其余向量线性表示。
(2)
α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs线性无关, α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs, β \beta β线性相关 ⇔ β \Leftrightarrow \beta ⇔β
可以由 α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs唯一线性表示。
(3) β \beta β可以由 α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs线性表示
⇔ r ( α 1 , α 2 , ⋯ , α s ) = r ( α 1 , α 2 , ⋯ , α s , β ) \Leftrightarrow r(\alpha_{1},\alpha_{2},\cdots,\alpha_{s}) = r(\alpha_{1},\alpha_{2},\cdots,\alpha_{s},\beta) ⇔r(α1,α2,⋯,αs)=r(α1,α2,⋯,αs,β)
4.向量组的秩与矩阵的秩之间的关系
设 r ( A m × n ) = r r(A_{m \times n}) = r r(Am×n)=r,则 A A A的秩 r ( A ) r(A) r(A)与 A A A的行列向量组的线性相关性关系为:
(1) 若 r ( A m × n ) = r = m r(A_{m \times n}) = r = m r(Am×n)=r=m,则 A A A的行向量组线性无关。
(2) 若 r ( A m × n ) = r < m r(A_{m \times n}) = r < m r(Am×n)=r<m,则 A A A的行向量组线性相关。
(3) 若 r ( A m × n ) = r = n r(A_{m \times n}) = r = n r(Am×n)=r=n,则 A A A的列向量组线性无关。
(4) 若 r ( A m × n ) = r < n r(A_{m \times n}) = r < n r(Am×n)=r<n,则 A A A的列向量组线性相关。
5. n \mathbf{n} n**维向量空间的基变换公式及过渡矩阵 **
若 α 1 , α 2 , ⋯ , α n \alpha_{1},\alpha_{2},\cdots,\alpha_{n} α1,α2,⋯,αn与 β 1 , β 2 , ⋯ , β n \beta_{1},\beta_{2},\cdots,\beta_{n} β1,β2,⋯,βn是向量空间 V V V的两组基,则基变换公式为:
( β 1 , β 2 , ⋯ , β n ) = ( α 1 , α 2 , ⋯ , α n ) [ c 11 c 12 ⋯ c 1 n c 21 c 22 ⋯ c 2 n ⋯ ⋯ ⋯ ⋯ ⋯ c n 1 c n 2 ⋯ c nn ] = ( α 1 , α 2 , ⋯ , α n ) C (\beta_{1},\beta_{2},\cdots,\beta_{n}) = (\alpha_{1},\alpha_{2},\cdots,\alpha_{n})\begin{bmatrix} & c_{11}\quad c_{12}\quad\cdots\quad c_{1n} \\ & c_{21}\quad c_{22}\quad\cdots\quad c_{2n} \\ & \quad\cdots\cdots\cdots\cdots\cdots \\ & c_{n1}\quad c_{n2}\quad\cdots\quad c_{\text{nn}} \\ \end{bmatrix} = (\alpha_{1},\alpha_{2},\cdots,\alpha_{n})C (β1,β2,⋯,βn)=(α1,α2,⋯,αn) c11c12⋯c1nc21c22⋯c2n⋯⋯⋯⋯⋯cn1cn2⋯cnn =(α1,α2,⋯,αn)C
其中 C C C是可逆矩阵,称为由基 α 1 , α 2 , ⋯ , α n \alpha_{1},\alpha_{2},\cdots,\alpha_{n} α1,α2,⋯,αn到基 β 1 , β 2 , ⋯ , β n \beta_{1},\beta_{2},\cdots,\beta_{n} β1,β2,⋯,βn的过渡矩阵。
**6.坐标变换公式 **
若向量 γ \gamma γ在基 α 1 , α 2 , ⋯ , α n \alpha_{1},\alpha_{2},\cdots,\alpha_{n} α1,α2,⋯,αn与基 β 1 , β 2 , ⋯ , β n \beta_{1},\beta_{2},\cdots,\beta_{n} β1,β2,⋯,βn的坐标分别是
X = ( x 1 , x 2 , ⋯ , x n ) T X = {(x_{1},x_{2},\cdots,x_{n})}^{T} X=(x1,x2,⋯,xn)T,
Y = ( y 1 , y 2 , ⋯ , y n ) T Y = \left( y_{1},y_{2},\cdots,y_{n} \right)^{T} Y=(y1,y2,⋯,yn)T 即:
γ = x 1 α 1 + x 2 α 2 + ⋯ + x n α n = y 1 β 1 + y 2 β 2 + ⋯ + y n β n \gamma = x_{1}\alpha_{1} + x_{2}\alpha_{2} + \cdots + x_{n}\alpha_{n} = y_{1}\beta_{1} + y_{2}\beta_{2} + \cdots + y_{n}\beta_{n} γ=x1α1+x2α2+⋯+xnαn=y1β1+y2β2+⋯+ynβn,则向量坐标变换公式为 X = C Y X = CY X=CY
或 Y = C − 1 X Y = C^{- 1}X Y=C−1X
,其中 C C C是从基 α 1 , α 2 , ⋯ , α n \alpha_{1},\alpha_{2},\cdots,\alpha_{n} α1,α2,⋯,αn到基 β 1 , β 2 , ⋯ , β n \beta_{1},\beta_{2},\cdots,\beta_{n} β1,β2,⋯,βn的过渡矩阵。
7.向量的内积
( α , β ) = a 1 b 1 + a 2 b 2 + ⋯ + a n b n = α T β = β T α (\alpha,\beta) = a_{1}b_{1} + a_{2}b_{2} + \cdots + a_{n}b_{n} = \alpha^{T}\beta = \beta^{T}\alpha (α,β)=a1b1+a2b2+⋯+anbn=αTβ=βTα
8.Schmidt正交化
若 α 1 , α 2 , ⋯ , α s \alpha_{1},\alpha_{2},\cdots,\alpha_{s} α1,α2,⋯,αs线性无关,则可构造 β 1 , β 2 , ⋯ , β s \beta_{1},\beta_{2},\cdots,\beta_{s} β1,β2,⋯,βs使其两两正交,且 β i \beta_{i} βi仅是 α 1 , α 2 , ⋯ , α i \alpha_{1},\alpha_{2},\cdots,\alpha_{i} α1,α2,⋯,αi的线性组合 ( i = 1 , 2 , ⋯ , n ) (i = 1,2,\cdots,n) (i=1,2,⋯,n),再把 β i \beta_{i} βi单位化,记 γ i = β i ∣ β i ∣ \gamma_{i} = \frac{\beta_{i}}{\left| \beta_{i} \right|} γi=∣βi∣βi,则 γ 1 , γ 2 , ⋯ , γ i \gamma_{1},\gamma_{2},\cdots,\gamma_{i} γ1,γ2,⋯,γi是规范正交向量组。其中
β 1 = α 1 \beta_{1} = \alpha_{1} β1=α1,
β 2 = α 2 − ( α 2 , β 1 ) ( β 1 , β 1 ) β 1 \beta_{2} = \alpha_{2} - \frac{(\alpha_{2},\beta_{1})}{(\beta_{1},\beta_{1})}\beta_{1} β2=α2−(β1,β1)(α2,β1)β1
,
β 3 = α 3 − ( α 3 , β 1 ) ( β 1 , β 1 ) β 1 − ( α 3 , β 2 ) ( β 2 , β 2 ) β 2 \beta_{3} = \alpha_{3} - \frac{(\alpha_{3},\beta_{1})}{(\beta_{1},\beta_{1})}\beta_{1} - \frac{(\alpha_{3},\beta_{2})}{(\beta_{2},\beta_{2})}\beta_{2} β3=α3−(β1,β1)(α3,β1)β1−(β2,β2)(α3,β2)β2
,
…
β s = α s − ( α s , β 1 ) ( β 1 , β 1 ) β 1 − ( α s , β 2 ) ( β 2 , β 2 ) β 2 − ⋯ − ( α s , β s − 1 ) ( β s − 1 , β s − 1 ) β s − 1 \beta_{s} = \alpha_{s} - \frac{(\alpha_{s},\beta_{1})}{(\beta_{1},\beta_{1})}\beta_{1} - \frac{(\alpha_{s},\beta_{2})}{(\beta_{2},\beta_{2})}\beta_{2} - \cdots - \frac{(\alpha_{s},\beta_{s - 1})}{(\beta_{s - 1},\beta_{s - 1})}\beta_{s - 1} βs=αs−(β1,β1)(αs,β1)β1−(β2,β2)(αs,β2)β2−⋯−(βs−1,βs−1)(αs,βs−1)βs−1
**9.正交基及规范正交基 **
向量空间一组基中的向量如果两两正交,就称为正交基;若正交基中每个向量都是单位向量,就称其为规范正交基。
线性方程组
1.克莱姆法则
线性方程组$\left{ \begin{matrix}
& a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} = b_{1} \
& a_{21}x_{1} + a_{22}x_{2} + \cdots + a_{2n}x_{n} = b_{2} \
& \quad\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \
& a_{n1}x_{1} + a_{n2}x_{2} + \cdots + a_{\text{nn}}x_{n} = b_{n} \
\end{matrix} \right.\ ,如果系数行列式 ,如果系数行列式 ,如果系数行列式D = \left| A \right| \neq 0$,则方程组有唯一解,
x 1 = D 1 D , x 2 = D 2 D , ⋯ , x n = D n D x_{1} = \frac{D_{1}}{D},x_{2} = \frac{D_{2}}{D},\cdots,x_{n} = \frac{D_{n}}{D} x1=DD1,x2=DD2,⋯,xn=DDn,其中 D j D_{j} Dj是把 D D D中第 j j j列元素换成方程组右端的常数列所得的行列式。
2.
n n n阶矩阵 A A A可逆 ⇔ A x = 0 \Leftrightarrow Ax = 0 ⇔Ax=0只有零解。 ⇔ ∀ b , A x = b \Leftrightarrow \forall b,Ax = b ⇔∀b,Ax=b总有唯一解,一般地, r ( A m × n ) = n ⇔ A x = 0 r(A_{m \times n}) = n \Leftrightarrow Ax = 0 r(Am×n)=n⇔Ax=0只有零解。
3.非奇次线性方程组有解的充分必要条件,线性方程组解的性质和解的结构
(1)
设 A A A为 m × n m \times n m×n矩阵,若 r ( A m × n ) = m r(A_{m \times n}) = m r(Am×n)=m,则对 A x = b Ax = b Ax=b而言必有 r ( A ) = r ( A ⋮ b ) = m r(A) = r(A \vdots b) = m r(A)=r(A⋮b)=m,从而 A x = b Ax = b Ax=b有解。
(2)
设 x 1 , x 2 , ⋯ x s x_{1},x_{2},\cdots x_{s} x1,x2,⋯xs为 A x = b Ax = b Ax=b的解,则 k 1 x 1 + k 2 x 2 + ⋯ + k s x s k_{1}x_{1} + k_{2}x_{2} + \cdots + k_{s}x_{s} k1x1+k2x2+⋯+ksxs当 k 1 + k 2 + ⋯ + k s = 1 k_{1} + k_{2} + \cdots + k_{s} = 1 k1+k2+⋯+ks=1时仍为 A x = b Ax = b Ax=b的解;但当 k 1 + k 2 + ⋯ + k s = 0 k_{1} + k_{2} + \cdots + k_{s} = 0 k1+k2+⋯+ks=0时,则为 A x = 0 Ax = 0 Ax=0的解。特别 x 1 + x 2 2 \frac{x_{1} + x_{2}}{2} 2x1+x2为 A x = b Ax = b Ax=b的解; 2 x 3 − ( x 1 + x 2 ) 2x_{3} - (x_{1} + x_{2}) 2x3−(x1+x2)为 A x = 0 Ax = 0 Ax=0的解。
(3)
非齐次线性方程组 Ax = b \text{Ax} = b Ax=b无解 ⇔ r ( A ) + 1 = r ( A ‾ ) ⇔ b \Leftrightarrow r(A) + 1 = r(\overline{A}) \Leftrightarrow b ⇔r(A)+1=r(A)⇔b不能由 A A A的列向量 α 1 , α 2 , ⋯ , α n \alpha_{1},\alpha_{2},\cdots,\alpha_{n} α1,α2,⋯,αn线性表示。
**4.奇次线性方程组的基础解系和通解,解空间,非奇次线性方程组的通解 **
(1)
齐次方程组 Ax = 0 \text{Ax} = 0 Ax=0恒有解(必有零解)。当有非零解时,由于解向量的任意线性组合仍是该齐次方程组的解向量,因此 Ax = 0 \text{Ax} = 0 Ax=0的全体解向量构成一个向量空间,称为该方程组的解空间,解空间的维数是 n − r ( A ) n - r(A) n−r(A),解空间的一组基称为齐次方程组的基础解系。
(2) η 1 , η 2 , ⋯ , η t \eta_{1},\eta_{2},\cdots,\eta_{t} η1,η2,⋯,ηt是 Ax = 0 \text{Ax} = 0 Ax=0的基础解系,即:
1) η 1 , η 2 , ⋯ , η t \eta_{1},\eta_{2},\cdots,\eta_{t} η1,η2,⋯,ηt是 Ax = 0 \text{Ax} = 0 Ax=0的解;
2) η 1 , η 2 , ⋯ , η t \eta_{1},\eta_{2},\cdots,\eta_{t} η1,η2,⋯,ηt线性无关;
Ax = 0 \text{Ax} = 0 Ax=0的任一解都可以由 η 1 , η 2 , ⋯ , η t \eta_{1},\eta_{2},\cdots,\eta_{t} η1,η2,⋯,ηt线性表出.
k 1 η 1 + k 2 η 2 + ⋯ + k t η t k_{1}\eta_{1} + k_{2}\eta_{2} + \cdots + k_{t}\eta_{t} k1η1+k2η2+⋯+ktηt是 Ax = 0 \text{Ax} = 0 Ax=0的通解,其中 k 1 , k 2 , ⋯ , k t k_{1},k_{2},\cdots,k_{t} k1,k2,⋯,kt是任意常数。