User:Braveorca/Algebra

This is a list of theorems and formulae from the books on algebra I'm studying. I'm putting them here because of Wikipedia's handy png-rendering capabilities. It's not in the main namespace because I feel that it isn't really appropriate, as many of the comments are my additional notes and the such. Once it's reasonably complete, I'll clean it up and it may be incorporated into an article somewhere, or possibly even a reference for anyone to use. Feel free to take/modify as you please (this is a wiki, after all).

Vectors

Basic operations

For any vectors $u,v,w\in \mathbb {R} ^{n}$ and any scalars $m,n\in \mathbb {R}$, the following properties hold:

• $(u+v)+w=u+(v+w)$ (associativity)
• $(u+0)=u$ (additive identity)
• $u+(-u)=0$ (additive inverse)
• $(u+v)=(v+u)$ (commutativity)
• $m(u+v)=mu+mv$ (left distributivity)
• $(m+n)u=mu+nu$ (right distributivity)
• $(mn)u=m(nu)$ (associativity)
• $1u=u$ (multiplicative identity)

Dot product

Also called inner product or scalar product. This, in contrast to the cross product, is a scalar, not a vector. Should this quantity equal zero, the vectors in quetion are orthogonal (perpendicular).

Given

$\mathbf {u} =(a_{1},a_{2},\cdots ,a_{n})={\begin{bmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{bmatrix}}$

and

$\mathbf {v} =(b_{1},b_{2},\cdots ,b_{n})={\begin{bmatrix}b_{1}\\b_{2}\\\vdots \\b_{n}\end{bmatrix}}$

we have

$\mathbf {u} \cdot \mathbf {v} =a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}$

Norm

Also called length. A vector with norm 1 is called a unit vector.

Given

$\mathbf {u} =(a_{1},a_{2},\cdots ,a_{n})={\begin{bmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{bmatrix}}$

we have

${\begin{Vmatrix}\mathbf {u} \end{Vmatrix}}={\sqrt {\mathbf {u} \cdot \mathbf {u} }}={\sqrt {a_{1}^{2}+a_{2}^{2}+\cdots +a_{n}^{2}}}$

Normalizing

Generates a unique unit vector in the same direction as a given vector.

Given

$\mathbf {u} =(a_{1},a_{2},\cdots ,a_{n})={\begin{bmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{bmatrix}}$

we have

${\hat {\mathbf {u} }}={1 \over {\begin{Vmatrix}u\end{Vmatrix}}}\mathbf {u} ={\mathbf {u} \over {\begin{Vmatrix}u\end{Vmatrix}}}$