kevinlui's site

January 8

2.1 Vectors

A vector is a list of number with addition and scalar multiplication defined. Given vectors $u=(u_1, u_2, \ldots, u_n)\in \mathbb{R}^n$, $v=(v_1,v_2,\ldots,v_n)\in \mathbb{R}^n$ of equal dimension and a scalar $c\in \mathbb{R}$, we define

• addition: $u+v=(u_1+v_1, u_2+v_2, \ldots, u_n+v_n)$,

• scalar multiplication: $cu=(cu_1,cu_2,\ldots,cu_n)$.

go over the geometry in class. tail to tip, parallelogram

Let $a,b$ be scalars and $u,v,w\in \mathbb{R}^n$. Then

• $u+v=v+u$,

• $a(u+v)=au+av$,

• $(a+b)u=au+bu$,

• $(u+v)+w = u+(v+w)$,

• $a(bu)=(ab)u$,

• $u+(-u)=0$,

• $u+0=0+u=u$,

• $1u=u$.

Definition: The If $u_1,u_2,\ldots,u_m$ are vectors and $c_1,c_2,\ldots,c_m$ are scalars, then $$c_1u_1+c_2u_2+\ldots+c_mu_m$$ is a linear combination of $u_1,\ldots,u_m$. Note that the constants can be negative or zero.