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/a9fa0efc9a7b56ec6f160bfb5a9c41be47205087)
![{\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.
Add by adding entry by entry:
Multiplication by a is multiplication of all entries by a.
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