A B → {\displaystyle {\overrightarrow {AB}}} B C → {\displaystyle {\overrightarrow {BC}}} A B → + B C → := A C → {\displaystyle {\overrightarrow {AB}}+{\overrightarrow {BC}}:={\overrightarrow {AC}}}
λ ⋅ ( x y ) = ( λ ⋅ x λ ⋅ y ) {\displaystyle \lambda \cdot {\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}{\lambda \cdot x}\\{\lambda \cdot y}\end{pmatrix}}}
‖ ( x y z ) ‖ = x 2 + y 2 + z 2 {\displaystyle {\begin{Vmatrix}{\begin{pmatrix}x\\y\\z\end{pmatrix}}\end{Vmatrix}}={\sqrt {x^{2}+y^{2}+z^{2}}}}
( x 1 , x 2 , . . . , x n ) + ( y 1 , y 2 , . . . , y n ) = ( x 1 + y 1 , . . . , x n + y n ) {\displaystyle (x_{1},x_{2},...,x_{n})+(y_{1},y_{2},...,y_{n})=(x_{1}+y_{1},...,x_{n}+y_{n})\ }
L i n ( M ) = { V → | v → = λ 1 v → 1 + . . . + λ k v → k , λ 1 , . . . , λ k ∈ K , v → 1 , . . . , v → k ∈ M , k ∈ N } {\displaystyle Lin(M)=\{{\vec {V}}|{\vec {v}}=\lambda _{1}{\vec {v}}_{1}+...+\lambda _{k}{\vec {v}}_{k},\quad \lambda _{1},...,\lambda _{k}\in K,\quad {\vec {v}}_{1},...,{\vec {v}}_{k}\in M,\quad k\in \mathbb {N} \}}
{ v → 1 , . . . , v → k } {\displaystyle \{{\vec {v}}_{1},...,{\vec {v}}_{k}\}\ } 0 = λ 1 ⋅ v → 1 + . . . + λ k ⋅ v → k {\displaystyle 0=\lambda _{1}\cdot {\vec {v}}_{1}+...+\lambda _{k}\cdot {\vec {v}}_{k}} λ i ≠ 0 {\displaystyle \lambda _{i}\neq 0}
v → 1 = ( 3 , 2 , 1 ) , v → 2 = ( 0 , 3 , 1 ) , v → 3 = ( 6 , 1 , 1 ) {\displaystyle {\vec {v}}_{1}=(3,2,1),{\vec {v}}_{2}=(0,3,1),{\vec {v}}_{3}=(6,1,1)} 2 ⋅ v → 1 + ( − 1 ) ⋅ v → 2 + ( − 1 ) ⋅ v 3 = ( 0 , 0 , 0 ) {\displaystyle 2\cdot {\vec {v}}_{1}+(-1)\cdot {\vec {v}}_{2}+(-1)\cdot v_{3}=(0,0,0)}
v → 1 , . . . , v → k {\displaystyle {\vec {v}}_{1},...,{\vec {v}}_{k}} w → 1 , . . . , w → e {\displaystyle {\vec {w}}_{1},...,{\vec {w}}_{e}}
v → i {\displaystyle {\vec {v}}_{i}} w → j {\displaystyle {\vec {w}}_{j}} { v → 1 , . . . , v → i − 1 , w → j , v → i + 1 , . . . , v → n } {\displaystyle \{{\vec {v}}_{1},...,{\vec {v}}_{i-1},{\vec {w}}_{j},{\vec {v}}_{i+1},...,{\vec {v}}_{n}\}}
A = ( a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ a m 1 a m 2 ⋯ a m n ) = ( a i j ) {\displaystyle A={\begin{pmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &&&\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{pmatrix}}=(a_{ij})}
( a 11 ⋯ a 1 n ⋮ ⋮ a m 1 ⋯ a m n ) + ( b 11 ⋯ b 1 n ⋮ ⋮ b m 1 ⋯ b m n ) = ( a 11 + b 11 ⋯ a 1 n + b 1 n ⋮ ⋮ a m 1 + b m 1 ⋯ a m n + b m n ) {\displaystyle {\begin{pmatrix}a_{11}&\cdots &a_{1n}\\\vdots &&\vdots \\a_{m1}&\cdots &a_{mn}\end{pmatrix}}+{\begin{pmatrix}b_{11}&\cdots &b_{1n}\\\vdots &&\vdots \\b_{m1}&\cdots &b_{mn}\end{pmatrix}}={\begin{pmatrix}a_{11}+b_{11}&\cdots &a_{1n}+b_{1n}\\\vdots &&\vdots \\a_{m1}+b_{m1}&\cdots &a_{mn}+b_{mn}\end{pmatrix}}}
( a 11 ⋯ a 1 n ⋮ ⋮ a m 1 ⋯ a m n ) ⋅ ( b 1 ⋮ b n ) = ( a 11 ⋅ b 1 + a 12 ⋅ b 2 + ⋯ + a 1 n + b n ⋮ ⋯ ( Z e i l e ⋅ S p a l t e ) ) {\displaystyle {\begin{pmatrix}a_{11}&\cdots &a_{1n}\\\vdots &&\vdots \\a_{m1}&\cdots &a_{mn}\end{pmatrix}}\cdot {\begin{pmatrix}b_{1}\\\vdots \\b_{n}\end{pmatrix}}={\begin{pmatrix}a_{11}\cdot b_{1}+a_{12}\cdot b_{2}+\cdots +a_{1n}+b_{n}\\\vdots \\\cdots (Zeile\cdot Spalte)\end{pmatrix}}}
( a 11 ⋯ a 1 n ⋮ ⋮ a i 1 ⋯ a i n ⋮ ⋮ a m 1 ⋯ a m n ) ⋅ ( b 11 ⋯ b 1 j ⋯ b 1 r ⋮ b n 1 ⋯ b n j ⋯ b n r ) = ( c 11 ⋯ c 1 r ⋮ c i j ⋮ c m 1 ⋯ c m r ) ∈ M ( m x r , k ) {\displaystyle {\begin{pmatrix}a_{11}&\cdots &a_{1n}\\\vdots &&\vdots \\a_{i1}&\cdots &a_{in}\\\vdots &&\vdots \\a_{m1}&\cdots &a_{mn}\end{pmatrix}}\cdot {\begin{pmatrix}b_{11}&\cdots &b_{1j}&\cdots &b_{1r}\\&&\vdots &\\b_{n1}&\cdots &b_{nj}&\cdots &b_{nr}\end{pmatrix}}={\begin{pmatrix}c_{11}&\cdots &c_{1r}\\\vdots &c_{ij}&\vdots \\c_{m1}&\cdots &c_{mr}\end{pmatrix}}\in M(mxr,k)}
c i j = a i 1 ⋅ b 1 j + a i 2 ⋅ b 2 j + ⋯ + a i n ⋅ b n j = ∑ k = 1 n a i k ⋅ b k j {\displaystyle c_{ij}=a_{i1}\cdot b_{1j}+a_{i2}\cdot b_{2j}+\dots +a_{in}\cdot b_{nj}=\sum _{k=1}^{n}a_{ik}\cdot b_{kj}}
( a 11 0 ⋱ ∗ a r r 0 0 0 0 ) a 11 ≠ 0 a 22 ≠ 0 ⋮ a r r ≠ 0 {\displaystyle {\begin{pmatrix}a_{11}&&&\\0&\ddots &&*\\&&a_{rr}\\0&0&0&0\end{pmatrix}}\quad {\begin{matrix}a_{11}\neq 0\\a_{22}\neq 0\\\vdots \\a_{rr}\neq 0\end{matrix}}}
( a 11 ′ b 1 ′ 0 ⋱ ∗ ⋮ 0 a r r ′ b r ′ 0 b r + 1 ′ 0 ⋮ b m ′ ) {\displaystyle {\begin{pmatrix}a'_{11}&&&b'_{1}\\0&\ddots &*&\vdots \\&0&a'_{rr}&b'_{r}\\&&0&b'_{r+1}\\&0&&\vdots \\&&&b'_{m}\end{pmatrix}}}
( a 11 ′ ⋮ | b 1 ′ 0 ⋱ T ⋮ S | ⋮ 0 a r r ′ ⋮ | b r ′ ) {\displaystyle {\begin{pmatrix}a'_{11}&&&\vdots &&|&b'_{1}\\0&\ddots &T&\vdots &S&|&\vdots \\&0&a'_{rr}&\vdots &&|&b'_{r}\\\end{pmatrix}}}
A ~ {\displaystyle {\tilde {A}}} A − 1 = 1 d e t A ⋅ A ~ {\displaystyle A^{-1}={\frac {1}{detA}}\cdot {\tilde {A}}} A ~ = ( a i j ~ ) {\displaystyle {\tilde {A}}=({\tilde {a_{ij}}})} a i j ~ = ( − 1 ) i + j ⋅ d e t A j i {\displaystyle {\tilde {a_{ij}}}=(-1)^{i+j}\cdot detA_{ji}}
A ⋅ x → = y → ⇔ A − 1 ⋅ y → = x → {\displaystyle A\cdot {\vec {x}}={\vec {y}}\Leftrightarrow A^{-1}\cdot {\vec {y}}={\vec {x}}}
∫ 0 1 f ( x ) g ( x ) d x {\displaystyle \int _{0}^{1}f(x)g(x)\mathrm {d} x}
| | v → | | = < v → , v → > {\displaystyle ||{\vec {v}}||={\sqrt {<{\vec {v}},{\vec {v}}>}}}