User:Braveorca/Algebra

This is a list of theorems from the books on linear algebra I'm studying. I'm putting them here because of Wikipedia's handy png-rendering capabilities. In time, this may be incorporated into an article somewhere.

Vectors

Basic operations

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

• ${\displaystyle (u+v)+w=u+(v+w)}$  (associativity)
• ${\displaystyle (u+0)=u}$  (additive identity)
• ${\displaystyle u+(-u)=0}$  (additive inverse)
• ${\displaystyle (u+v)=(v+u)}$  (commutativity)
• ${\displaystyle k(u+v)=ku+kv}$ 
• ${\displaystyle ===Dotproduct===Alsocalledinnerproduct.Shouldthisquantityequalzero,thevectorsinquetionareorthogonal(perpendicular).Given<math>\mathbf {u} =(a_{1},a_{2},\cdots ,a_{n})={\begin{bmatrix}a_{1}\\a_{2}\\\vdots \\a_{n}\end{bmatrix}}}$ 

and

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

we have

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

Norm

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

Given

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

we have

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

Normalizing

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

Given

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

we have

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