Vector space

vector space의 조건

  1. There exist an additive identity $0$ (영벡터가 존재)
  2. For each $x \in V$, there exists an additive inverse $-x$ (역벡터 존재)
  3. There exist a multiplicative identitiy in $\mathbb{R}$ such that $1x = x$ for all $x \in V$
  4. Commutativity(교환 법칙): $x + y = y + x$ for all $x, y \in V$
  5. Associativity(결합 법칙): $(x + y) + z = x + (y+z)$ and $\alpha(\beta x) = (\alpha\beta)x$ for all $x, y, z\in V$ and $\alpha,\beta \in \mathbb{R}$
  6. Distributivity(분배법칙): $\alpha (x+y)=\alpha x+\alpha y$ and $(\alpha + \beta)x = \alpha x +\beta x$ for all $x, y, z\in V$ and $\alpha,\beta \in \mathbb{R}$

sparsity

span

Superposition

superposition(linear function)

$f: R^n \rightarrow R$ satisfies superposition property if

$$ f(\alpha x+\beta y)=\alpha f(x) + \beta f(y) $$

→ f is a linear function!

inner product function

Affine function

a function that is linear plus a constant

$$ f(x) = a^Tx+b $$

Untitled

affine 함수는 $\alpha + \beta = 1$일 때만 $f(\alpha x+\beta y)=\alpha f(x) + \beta f(y)$을 만족한다. ($\alpha f(x) + \beta f(y)$가 직선 위에 있을 때 (내분점, 외분점))

Norm

Euclidean norm