What "finitely generated" vector space means? [duplicate]
Can someone please help me understand the concept of "finitely generated" vector space, I am having a hard time visualizing and understanding it..
Thanks :)
$\endgroup$ 21 Answer
$\begingroup$Nothing does it as an example and some intuition. First of all: What is a vector space?
I like to think of it as a space $(V,+,\cdot)$ of objects $v \in V$ that can be added with each other ($v+u \in V$) and scaled (i.e. scalar multiplicated $\lambda \cdot v=\lambda v \in V$) by some $\lambda \in \mathbb R$ (let's just assume we have a real vector space).
If you now think of $\mathbb R ^3$ as a vector space, which describes our real world, you can reach every point in space by using 3 different directions (up/down left/right back/forth) and add and scale them approriately.
For instance pick $ \space\left(\begin{array}{c} 1\\3\\-2\end{array}\right) \in \mathbb R ^3$. Now if you pick the three vectors $\space\left(\begin{array}{c} 1\\0\\0\end{array}\right), \space\left(\begin{array}{c} 0\\1\\0\end{array}\right),\space\left(\begin{array}{c} 0\\0\\1\end{array}\right)$, you can obtain your first vector by summing and scaling i.e:
$$ 1 \cdot \space\left(\begin{array}{c} 1\\0\\0\end{array}\right)+ 3 \cdot\space\left(\begin{array}{c} 0\\1\\0\end{array}\right)+ (-2)\cdot\space\left(\begin{array}{c} 0\\0\\1\end{array}\right)=\space\left(\begin{array}{c} 1\\3\\-2\end{array}\right)$$
So what have we done? We picked a vector in our vector space and were able to express it in terms of finitely many vectors in our vector space only requiring $(+,\cdot)$. If one wants to be rigorous (which I urgue you to try!) one should abstract this idea to general definitions of:
- Basis of a vector space
- the span
- linear indipendence
- the dimension of a vector space
All these definitions are very important (look them up if you don't know them or revisit them otherwise) and indispensable for understanding what a vector space is. You can look them up and try to match them with the example I just gave as a motivation why one defines things as they are defined. Feel free to ask if something is unclear.
$\endgroup$ 2
