Vectors

• A vector v is a set of numbers
• Components: the entries of a vector.
  • The i-th component of vector v refers to $v_i$
  • $v_1 = 1, v_2 = 2, v_3 = 3$
• If a vector only has less than for components, you can visualize it.

Vector Set

$\begin{Bmatrix} \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}, \begin{bmatrix} 6 \\ 8 \\ 9 \end{bmatrix}, \begin{bmatrix} 9 \\ 0 \\ 2 \end{bmatrix} \end{Bmatrix}$

• A vector set can contain infinite elements $L = \begin{Bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} : x_1 + x_2 = 1 \end{Bmatrix}$

$\begin{bmatrix} 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0.3 \\ 0.7 \end{bmatrix}, \begin{bmatrix} 0.7 \\ 0.3 \end{bmatrix}$

• $R^n$: We denote the set of all vectors with n entries by $R^n$

Scalar Multipication

$v = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}$

$cv = \begin{bmatrix} cv_1 \\ cv_2 \end{bmatrix}$

Vector Addition

$(a_1+b_1,a_2+b_2)$

Properties of Vector

For any vectors $u$,$v$ and $w$ in $R^n$, and any scalars a and b

• $u + v = v + u$
• $(u+v)+w = u + (v+w)$
• There is an element $0$ in $R^n$ such that $0 + u=u$
• There is an element $u'$ in $R^n$ such that $u'+u=0$
  • $u'= -u$
  • $0 = \begin{bmatrix} 0 \\ ... \\ 0 \end{bmatrix}$ zero vector
• $1u = u$
• $(ab)u=a(bu)$
• $a(u+v)=au+av$
• $(a+b)u=au+bu$

PS: The objects have the following 8 properties ar "vectors"