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  1. Linear Algebra
  2. Vectors and Spaces

Linear combinations and spans

PreviousVectorsNextLinear dependence and independence

Last updated 5 years ago

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向量加上任意 實數 scalar 後，並且透過加法組合在一起時，產生一個 線性 的組合即為 Linear combination

a1v1⃗+a2v2⃗+⋯+anvn⃗.∣an∈Ra_1\vec{v_1} + a_2\vec{v_2}+ \cdots + a_n\vec{v_n}. | a_n \in \mathbb{R}a1​v1​​+a2​v2​​+⋯+an​vn​​.∣an​∈R

舉個例子，以下兩個向量經過與 scalar 相乘後，可以組合並且表示一個新的向量

3[12]+2[03]=[312]3 \begin{bmatrix} 1\\ 2\end{bmatrix} +2 \begin{bmatrix} 0\\ 3\end{bmatrix}= \begin{bmatrix} 3\\ 12\end{bmatrix}3[12​]+2[03​]=[312​]

因為 Vectors 可以與任意實數相乘，產生的 linear combination 就可以任意表示其他向量，這個現象叫作 Span

例如以下兩個向量不管 a1 和 a2 為何，在 combine 之後只能 span 這兩條向量原本的那條線

a1[12]+a2[24]=a1[12]+2a2[12]=a1+2a2[12]=a[12]\begin{aligned} a_1 \begin{bmatrix} 1\\ 2\end{bmatrix}+ a_2\begin{bmatrix} 2\\ 4\end{bmatrix}&=\\ a_1 \begin{bmatrix} 1\\ 2\end{bmatrix}+ 2a_2\begin{bmatrix} 1\\2\end{bmatrix}&=\\ a_1 +2a_2 \begin{bmatrix} 1\\ 2\end{bmatrix}&= a\begin{bmatrix} 1\\ 2\end{bmatrix} \end{aligned}a1​[12​]+a2​[24​]a1​[12​]+2a2​[12​]a1​+2a2​[12​]​===a[12​]​

而底下兩個向量 (1,0) 和 (0,1) 卻可以 span 整個二維平面的任意兩點，我們可以這樣表示

v1⃗=(1,0),v2⃗=(0,1)a1v1⃗+a2v2⃗=R2\begin{aligned} &\vec{v_1} = (1,0), \vec{v_2} = (0,1)\\ &a_1\vec{v_1} + a_2\vec{v_2} = \mathbb{R}^2 \end{aligned}​v1​​=(1,0),v2​​=(0,1)a1​v1​​+a2​v2​​=R2​
https://youtu.be/Qm_OS-8COwU