Scientia
目錄
在整個線性代數中，最基本的單位就是一個向量，假設一個擁有兩個值的向量像是
![\[V = \begin{bmatrix}v_{1} \\ v_{2} \end{bmatrix}\]](https://scientia-potentia-est.com/wp-content/ql-cache/quicklatex.com-57da565653cb2b303d15e4deea34730a_l3.png "Rendered by QuickLaTeX.com")
向量加法
假設另一組向量
一樣包含於兩個值
可以進行向量的加法像是
![\[v + w = \begin{bmatrix} v_{1} \\ v_{2} \end{bmatrix} + \begin{bmatrix} w_{1} \\ w_{2} \end{bmatrix} = \begin{bmatrix} v_{1} + w_{1} \\ v_{2} + w_{2}\end{bmatrix}\]](https://scientia-potentia-est.com/wp-content/ql-cache/quicklatex.com-1c2de62580c8452a3145385a156e1416_l3.png "Rendered by QuickLaTeX.com")
向量乘法
同樣的可以將向量乘上一組實數
![\[2 v = 2 \begin{bmatrix} v_{1} \\ v_{2}\end{bmatrix} = \begin{bmatrix} 2v_{1} \\ 2v_{2}\end{bmatrix}\]](https://scientia-potentia-est.com/wp-content/ql-cache/quicklatex.com-3f63510198b0d8fc91ff4980949d7006_l3.png "Rendered by QuickLaTeX.com")
![\[c\begin{bmatrix}v_{1} \\ v_{2}\end{bmatrix} = \begin{bmatrix} cv_{1} \\ cv_{2} \end{bmatrix}, where \> c \> \in \mathbb{Z}\]](https://scientia-potentia-est.com/wp-content/ql-cache/quicklatex.com-37fb1e069d09edcfa6f30835498b6b5b_l3.png "Rendered by QuickLaTeX.com")
在實數的運算下加法跟乘法一樣可以同時計算
![\[cv + dw = \begin{bmatrix} cv_{1}+dw_{1} \\ cv_{2}+dw_{2}\end{bmatrix}\]](https://scientia-potentia-est.com/wp-content/ql-cache/quicklatex.com-b3e3265010d0da95f0fd721aabc891af_l3.png "Rendered by QuickLaTeX.com")
這被稱作
跟的線性組合，其中
跟
是實數，而不斷擴充下去可以到
個維度的線性組合
線性方程式
傳統上可以看到聯立方程式如下
![\[\begin{cases} x-2y = 1 \\ 3x+2y = 11\end{cases}\]](https://scientia-potentia-est.com/wp-content/ql-cache/quicklatex.com-6350698d47e0fe4fca4e238a6273c1bb_l3.png "Rendered by QuickLaTeX.com")
可以使用矩陣的形式來表示一組聯立方程式像是
![\[\begin{bmatrix}1 & -2 \\ 3 & 2 \end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}1 \\ 11\end{bmatrix}\]](https://scientia-potentia-est.com/wp-content/ql-cache/quicklatex.com-c6c2daa352c94f532da39335f5d3fb8f_l3.png "Rendered by QuickLaTeX.com")
可以將其視為一組
的矩陣，另外通常會表示成
![\[Ax = b\]](https://scientia-potentia-est.com/wp-content/ql-cache/quicklatex.com-7e81ccbb9d5f42ca099727b9b1632bfe_l3.png "Rendered by QuickLaTeX.com")
是一組矩陣，目前是用row的角度在觀察這一組線性方程式，如果換成column的角度觀察的話可以表示成：
![\[x \begin{bmatrix}1 \\ 3 \end{bmatrix} + y\begin{bmatrix} -2 \\ 2\end{bmatrix} = \begin{bmatrix}1 \\ 11 \end{bmatrix}\]](https://scientia-potentia-est.com/wp-content/ql-cache/quicklatex.com-0b36d2ae32347aac49696e64d6a059ab_l3.png "Rendered by QuickLaTeX.com")
我們的目標就是找出左邊矩陣可能的線性組合來滿足右邊的矩陣，
![\[3 \begin{bmatrix}1 \\ 3 \end{bmatrix} + 1\begin{bmatrix} -2 \\ 2\end{bmatrix} = \begin{bmatrix}1 \\ 11 \end{bmatrix}\]](https://scientia-potentia-est.com/wp-content/ql-cache/quicklatex.com-f1f342f53aef50b9619d961913da4418_l3.png "Rendered by QuickLaTeX.com")
同樣的可以將這樣的表示法擴展到個維度上。
![\[\begin{cases}2x + 4y - 2z = 2 \\ 4x + 9y - 3z = 8 \\ -2x - 3y + 7z = 10\end{cases}\]](https://scientia-potentia-est.com/wp-content/ql-cache/quicklatex.com-1cb9b801b3ee296475066aa02ad581d8_l3.png "Rendered by QuickLaTeX.com")
