Linear Algebra

1. tuples→ signals
2. linear functions on tuples → systems

Tuples and Equations

Name and Notation

• “n-tuple” for an array $[a_1,a_2 \dots , a_n]$ or $\begin{bmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{bmatrix}$ instead of “vector of length n” or “n-dimensional vector”
  • As direction and magnitude is not always defined

Functions

• Domain → 정의역
• Range → 치역

1. Every member in the domain has to be mapped
2. Every mapping must be unique

onto functions (전사함수)

Functions whose Codomain = Range

1-1 functions (일대일 대응 함수)

$$ \text{Two different numbers must be mapped into different values } \\ a \neq b \ \text{in Domain} \Rightarrow f(a) \neq f(b)\ \text{in Range} \\ \text{or} \\ f(a) = f(b) \Rightarrow a=b $$

Linear functions on tuples

• A linear function preserves the addition from domain to range
  • $f(x) = 5x$ is a linear function from $R$ to $R$
  • $f(a+b) = f(a) + f(b)$ for any real $a$ and $b$
  • $f(a+b)$ : addition in domain → send the sum of $a$ and $b$ to the range by $f$
  • $f(a) + f(b)$ : addition in range → send $a$ and $b$ individually to the range by $f$ and add
  • $g(x) = 3x + 1$ is an affine function from $R$ to $R$ (not linear)