Vectors in \(\mathbb{R}^2\) → 실수 2차원 공간
\(\mathbf{u} = \left[\begin{matrix}3 \\ -1 \\\end{matrix}\right], \mathbf{v} = \left[\begin{matrix}0.2 \\ 0.3 \\\end{matrix}\right], \mathbf{u} = \left[\begin{matrix}w_1 \\ w_2 \\\end{matrix}\right]\)
\(\vec{u} = (3, -1), \vec{v} = (0.2, 0.3), \vec{w} = (w_1, w_2)\)
Scalar Multiplication
백터 네 모든 값에 스칼라 값을 곱해준다
\(c = 5, \mathbf{u} = \left[\begin{matrix}3 \\ -1 \\\end{matrix}\right]\) 일 때,
\(c\mathbf{u} = 5 \left[\begin{matrix}3 \\ -1 \\\end{matrix}\right] = \left[\begin{matrix}5\cdot3 \\ 5\cdot(-1) \\\end{matrix}\right] = \left[\begin{matrix}15 \\ -5 \\\end{matrix}\right]\)
Vectors in \(\mathbb{R}^n\)
\(c\mathbf{u} = \left[\begin{matrix}u_1 \\ u_2 \\ \cdots \\ u_n\end{matrix}\right] \\ \mathbf{u} = (u_1, u_2, \cdots, u_n)\)
Algebraic Properties of \(\mathbb{R}^n\)
\(\mathbf{u, v, w} \mbox{ are } \mathbb{R}^n \mbox{ and } c d \mbox{ are scalar}\)
- u + v = v + u
- (u + v) + w = u + (v + w)
- u + 0 = 0 + u = u
- u + (-u) = -u + u = 0
- c(u + v) = cu + cv
- (c + d)u = cu + du
- c(du) = (cd)u
- 1u = u
Linear Combinations
Linear combination of \(\mathbf{v_1, v_2, \cdots, v_p} \mbox{ is } \mathbb{R}^n \mbox{with weights } c_1, c_2, \cdots, c_p \mbox{ is scalar}\)
y = \(c_1\mathbf{v_1} + c_2\mathbf{v_2} + \cdots + c_p\mathbf{v_p}\)
A vector equation
\(x_1\mathbf{a_1}+x_2\mathbf{a_2}+\cdots+x_n\mathbf{a_n} = \mathbf{b}\)
has the same solution set as the linear system whose augmented matrix is
\(\left[\mathbf{a_1} \mathbf{a_2} \cdots \mathbf{a_n} \mathbf{b}\right]\)
→ 각 벡터를 집어넣은 Augmented Matrix 와 동일한 해를 지닌다.
Span{\(\mathbf{v}\)}
is the collection of all vectors that can be written in the form
\(c_1\mathbf{v_1} + c_2\mathbf{v_2} + \cdots + c_p\mathbf{v_p}\)