VECTORS in Rⁿ and VECTOR EQUATIONS

n = 2:

${\bf u}= \left[ \begin{array}{c}u_1 \\ u_2 \end{array} \right ] \in R^2$, a 2-vector. Addition ${\bf u}\ +{\bf v}= \left[\begin{array}{c}u_1+v_1\\ u_2+v_2\end{array}\right ]$,
scalar multiplication $c{\bf u}= \left[ \begin{array}{c}cu_1\\ cu_2\end{array} \right ]$, zero vector ${\bf0}= \left[ \begin{array}{c}0\\ 0\end{array} \right ]$.
Geometry and parallelogram rule

n = 3:

${\bf u}= \left[ \begin{array}{c}u_1\\ u_2\\ u_3\end{array} \right ] \in R^3$, a 3-vector. Addition ${\bf u}\ +{\bf v}=\left[\begin{array}{c}u_1+v_1\\ u_2+v_2\\ u_3+v_3\end{array}\right]$,
scalar multiplication $c{\bf u}= \left[ \begin{array}{c}cu_1\\ cu_2\\ cu_3 \end{array} \right ]$, zero vector ${\bf0}= \left[ \begin{array}{c}0\\ 0\\ 0\end{array} \right ]$.

General n:

${\bf u}= \left[ \begin{array}{c}u_1\\ \vdots\\ u_n\end{array} \right ] \in R^n$, an n-vector. Addition ${\bf u}\ +{\bf v}=\left[\begin{array}{c}u_1+v_1\\ \vdots\\ u_n+v_n\end{array}\right]$,
scalar multiplication $c{\bf u}= \left[\begin{array}{c}cu_1\\ \vdots \\ cu_n \end{array} \right]$, zero vector ${\bf0}= \left[ \begin{array}{c}0\\ \vdots \\ 0\end{array} \right]$.
Algebraic properties: for all ${\bf u}, {\bf v}, {\bf w}\in R^n$, and scalars c, d

1.

u + v = v + u

2.

( u + v ) + w = u + ( v + w )

3.

u + 0 = u

4.

u + (-u) = u - u= 0

5.

c(u + v) = cu + cv

6.

(c+d)u = cu + du

7.

c(du) = (cd)u

8.

1u = u

Linear combinations:

• linear combination of ${\bf v}_1, {\bf v}_2, \ldots, {\bf v}_p \in R^n$, using $c_1, c_2, \ldots, c_p \in R$, is
  
  $${\bf y}= c_1{\bf v}_1 + c_2{\bf v}_2 + \cdots + c_p{\bf v}_p .$$
  
  

• $Span\{{\bf v}_1, {\bf v}_2, \ldots, {\bf v}_p\}$ is the set of all possible linear combinations
  
  $$c_1{\bf v}_1 + c_2{\bf v}_2 + \cdots + c_p{\bf v}_p .$$
  
  
  Geometry: For ${\bf u}, {\bf v}\neq {\bf0}$, $Span\{{\bf u}\}$ is a line, $Span\{{\bf u}, {\bf v}\}$ is (usually) a plane.
  Vector equations: ${\bf b}\in Span\{{\bf a}_1,{\bf a}_2, \ldots,{\bf a}_n\}$ iff the linear system
  
  $$x_1{\bf a}_1 + x_2{\bf a}_2 + \cdots + x_p{\bf a}_p = {\bf b},$$
  
  
  is consistent.

The MATRIX EQUATION $A{\bf x}= {\bf b}$

Matrix-Vector Multiplication:

If A is an $m \times n$ matrix and ${\bf x}\in R^n$,

$$A{\bf x}= [ {\bf a}_1\ {\bf a}_2\ \ldots\ {\bf a}_n ] \left[... ...ight ] = x_1{\bf a}_1 + x_2{\bf a}_2 + \cdots + x_n{\bf a}_n$$

Linear Systems:

These representations all have the same solution set:

• $A{\bf x}= {\bf b}$,
• vector equation
  
  $$x_1{\bf a}_1 + x_2{\bf a}_2 + \cdots + x_n{\bf a}_n = {\bf b},$$
  
  

• linear system with augmented matrix
  
  $$\left[ \begin{array}{ccccc} a_{11} & a_{12} & \ldots & a_{1n}... ... a_{m1} & a_{m2} & \ldots & a_{mn} & b_m \end{array}\right].$$
  
  

Existence of solutions:

$A{\bf x}= {\bf b}$ has a solution iff ${\bf b}$ is a linear combination of ${\bf a}_1, {\bf a}_2, ..., {\bf a}_n$.

Computation of $A{\bf x}$:

• i^th component of $A{\bf x}$ is $(A{\bf x})_i=a_{i1}x_1+a_{i2}x_2+\cdots + a_{in}x_n$
• $A({\bf u}+{\bf v}) = A{\bf u}+ A{\bf v}$
• $A(c{\bf u}) = c(A{\bf u})$.

Identity Matrix:

• I has ones on the diagonal and zeros elsewhere.
• I_n is an $n\times n$ identity matrix.
  
  $$I_n = \left[ \begin{array}{cccc} 1 & 0 & \ldots & 0 \\ 0 & ... ... & \ldots & \vdots \\ 0 & 0 & \ldots & 1 \end{array}\right].$$
  
  

• $I_n{\bf x}= {\bf x}$ for all ${\bf x}\in R^n$.