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#<math> \forall x\; \exists\ y \ s.t.\ x+y=0 \ </math> |
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#<math> \forall x\; \exists\ y \ s.t.\ x+y=0 \ </math> |
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#<math> 1.x=x \ </math> |
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#<math> 1.x=x \ </math> |
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#<math> a(bx=(ab)x \ </math> |
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#<math> a(bx)=(ab)x \ </math> |
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#<math> a(x+y)=ax+ay \ </math> |
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#<math> a(x+y)=ax+ay \ </math> |
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#<math> (a+b)x=ax+bx \ </math> |
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#<math> (a+b)x=ax+bx \ </math> |
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====''Examples''==== |
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====''Examples''==== |
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'''Ex.1.''' |
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'''Ex.1.''' |
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<math> F^n= \big\{ (a_1,a_2,a_3,...,a_{n-1},a_n):\forall i\ a_i \in F \big\} </math> <br/> |
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<math> F^n= \lbrace(a_1,a_2,a_3,...,a_{n-1},a_n):\forall i\ a_i \in F \rbrace </math> <br/> |
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<math> n \in \mathbb{Z}\ , n \ge 0 </math> <br/> |
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<math> n \in \mathbb{Z}\ , n \ge 0 </math> <br/> |
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<math> x=(a_1,...,a_2)\ y=(b_1,...,b_2)\ </math> <br/> |
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<math> x=(a_1,...,a_2)\ y=(b_1,...,b_2)\ </math> <br/> |
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<math> a\in F\ ax=(aa_1,aa_2,...,aa_n) </math> <br/> |
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<math> a\in F\ ax=(aa_1,aa_2,...,aa_n) </math> <br/> |
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<math> In \ \mathbb{Q}^3 \ ( \frac{3}{2},-2,7)+( \frac{-3}{2}, \frac{1}{3},240)=(0, \frac{-5}{3},247) </math> <br/> |
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<math> In \ \mathbb{Q}^3 \ ( \frac{3}{2},-2,7)+( \frac{-3}{2}, \frac{1}{3},240)=(0, \frac{-5}{3},247) </math> <br/> |
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<math> 7( \frac{1}{5},\frac{1}{7},\frac{1}{9})=( \frac{7}{5},1,\frac{7}{9}) </math> <br/> |
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<math> 7\left( \frac{1}{5},\frac{1}{7},\frac{1}{9}\right)=\left( \frac{7}{5},1,\frac{7}{9}\right) </math> <br/> |
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'''Ex.2.''' |
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'''Ex.2.''' |
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<math> V=M_{m \times n}(F)=\Bigg\{\begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & |
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<math> V=M_{m\times n}(F)=\lbrace\begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & |
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& \vdots \\ a_{m1} & \cdots & a_{mn}\end{pmatrix}: a_{ij} \in F \Bigg\} </math> <br/> |
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& \vdots \\ a_{m1} & \cdots & a_{mn}\end{pmatrix}: a_{ij} \in F \rbrace </math> <br/> |
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<math> M_{3\times 2}( \mathbb{R})\ni \begin{pmatrix} 7 & -7 \\ \pi & \mathit{e} \\ -5 & 2 \end{pmatrix} </math> <br/> |
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<math> M_{3\times 2}( \mathbb{R})\ni \begin{pmatrix} 7 & -7 \\ \pi & \mathit{e} \\ -5 & 2 \end{pmatrix} </math> <br/> |
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Addition by adding entry by entry: |
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Add by adding entry by entry:<math> M_{2\times 2}\ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}+\begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}=\begin{pmatrix} {a_{11}+b_{11}} & {a_{12}+b_{12}} \\ {a_{21}+b_{21}} & {a_{22}+b_{22}} \end{pmatrix}</math> <br/> |
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Multiplication by a is multiplication of all entries by a. <br/> |
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<math> M_{2\times 2}\ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}+\begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}=\begin{pmatrix} {a_{11}+b_{11}} & {a_{12}+b_{12}} \\ {a_{21}+b_{21}} & {a_{22}+b_{22}} \end{pmatrix}</math> <br/> |
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Multiplication by multiplying scalar c to all entries by M. |
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<math> c\cdot M_{2\times 2}\ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}=\begin{pmatrix} c\cdot a_{11} & c\cdot a_{12} \\ c\cdot a_{21} & c\cdot a_{22} \end{pmatrix}</math> <br/> <br/> |
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Zero matrix has all entries = 0: |
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<math> 0_{M_{m\times n}}=\begin{pmatrix} 0 & \cdots & 0 \\ \vdots & |
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<math> 0_{M_{m\times n}}=\begin{pmatrix} 0 & \cdots & 0 \\ \vdots & |
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& \vdots \\ 0 & \cdots & 0\end{pmatrix} </math> <br/> |
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& \vdots \\ 0 & \cdots & 0\end{pmatrix} </math> <br/> |
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<math> f,g \in \mathcal{F}(S,\mathbb{R}) </math> <br/> |
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<math> f,g \in \mathcal{F}(S,\mathbb{R}) </math> <br/> |
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<math> (f+g)(t)=f(t)+g(t)\ for\ any\ t\in S </math> <br/> |
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<math> (f+g)(t)=f(t)+g(t)\ for\ any\ t\in S </math> <br/> |
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<math> (af)(t)=a.f(t)\ </math> |
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<math> (af)(t)=a\cdot f(t)\ </math> |
A force has a direction & a magnitude.
Force Vectors
- There is a special force vector called 0.
- They can be added.
- They can be multiplied by any scalar.
====Properties==== (convention: x,y,z-vectors; a,b,c-scalars)
![{\displaystyle x+y=y+x\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f85f5493a398e0abccc8e10703ab0ee683fddac5)
![{\displaystyle x+(y+z)=(x+y)+z\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf050bca647083605a3f536711f938c78f279146)
![{\displaystyle x+0=x\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/40c69b5b348f9ca362a4e695ff90059ba9402de9)
![{\displaystyle \forall x\;\exists \ y\ s.t.\ x+y=0\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/be1a13a1bd60948b8438d320eda63f5802b8d60a)
![{\displaystyle 1.x=x\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/17cf68c61fd09980f50b69c527ac8e740a095d1a)
![{\displaystyle a(bx)=(ab)x\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7523325fb2d65ffe394a7c931f85c2855765c4b3)
![{\displaystyle a(x+y)=ax+ay\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/db5d5a003933d5b90bfe76e3e2cb292316cc3452)
![{\displaystyle (a+b)x=ax+bx\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9a06e8cbd0b15dddc24c6d3343d9ead4c136f4b)
=====Definition===== Let F be a field "of scalars". A vector space over F is a set V (of "vectors") along with two operations:
![{\displaystyle +:V\times V\to V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/982bd4f2bd4daba2cb683b2ea4ac907d5422dcce)
, so that
![{\displaystyle \forall x,y\in V\ x+y=y+x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ebfcbab8f76165500ffe744313298026091e978)
![{\displaystyle \forall x,y\in V\ x+(y+z)=(x+y)+z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd1a9bbeb7293e94c72b05c3cc122b9f4652b87)
![{\displaystyle \exists \ 0\in Vs.t.\;\forall x\in V\ x+0=x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f62a3124291a3c4614dd490089b7822baa8bb5)
![{\displaystyle \forall x\in V\;\exists \ y\in V\ s.t.\ x+y=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be891d4519d5044e7619c7948eb77cb7380135fc)
![{\displaystyle 1.x=x\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/17cf68c61fd09980f50b69c527ac8e740a095d1a)
![{\displaystyle a(bx)=(ab)x\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7523325fb2d65ffe394a7c931f85c2855765c4b3)
![{\displaystyle a(x+y)=ax+ay\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/db5d5a003933d5b90bfe76e3e2cb292316cc3452)
![{\displaystyle \forall x\in V\ ,\forall a,b\in F\ (a+b)x=ax+bx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65d86e129b5d7dfc56c0e12e61e2001f04dff51f)
9.
Examples
Ex.1.
Ex.2.
Addition by adding entry by entry:
Multiplication by multiplying scalar c to all entries by M.
Zero matrix has all entries = 0:
Ex.3.
form a vector space over
.
Ex.4.
F is a vector space over itself.
Ex.5.
is a vector space over
.
Ex.6.
Let S be a set. Let