[Math] Vector Space – how to visualize it for understanding

vector-spaces

I read on Wikipedia about vector spaces, but I don't understand them in a way that I can visualize the vector spaces in my head. During the process of understanding, I had several concepts in my head and I am at a point now, where I am totally confused. Maybe I am in a dead end as well. I have drawn four of these concepts, so you can imagine what happened in my head.

Pictures: my approaches for vector spaces

Picture A
$ \vec{r} $ is the vector space, which means the space is linear on the line of the vector. $ \vec{r} $ contains infinite vectors like $ \vec{a} $, $ \vec{b} $ and $ \vec{c} $. The last three vectors only exist in $ \vec{r} $ or vector spaces which are bigger or equal to themselves. An orthogonal vector of $ \vec{b} $ is not a part of $ \vec{r} $.

Picture B
The vector space is an area where one or multiple vectors like $ \vec{r} $ and $ \vec{m} $ exist. The space is infinite, which doesn't make much sense to define a space. But it is a space. In the picture it is the striped zone of the diagram.

Picture C
$ \vec{r} $ can be build by the linear combination of $ \vec{a} $ + $ \vec{b} $, $ \vec{c} $ + $ \vec{d} $ or any other combination of two vectors within the red striped zone. But what is with combinations outside of the red striped zone? Here it destroys my concept probably.

Picture D
$ \vec{r} $ is the shortest vector to the target point. $ \vec{a} $, $ \vec{b} $, $ \vec{c} $ and $ \vec{d} $ are one linear combination of multiple possible linear combinations to the target. Is the red striped area the vector space or red and yellow together?

Is one of my concepts the right concept of vector spaces?

I really appreciate your inputs and hope to get a explanation which my brain can visualize. Maybe you could draw it?

Best Answer

The following are primary examples of vector spaces (over the real numbers):

  • A one point set, regarding the point as the origin, i.e. the zero vector $\{0\}$. This space is $0$ dimensional.

  • A full line through the origin (basically it's along the lines of your picture A, but we also consider negative and every multiples of its vectors). The lines are $1$ dimensional.

  • A full plane through the origin, including all its points. These are $2$ dimensional.

  • The physical 3d space you can consider as a $3$ dimensional vector space after fixing a point for origin: you can add vectors and multiply them by real numbers: that's what the abstract definition says.

We can observe that in all these geometric examples, the elements of the given set can be coordinatized by base vectors, namely we have to fix exactly as many base vectors as the given 'dimension'.
This, on one hand, means that the elements of the given set can be represented by a single coordinate (for a line) / a pair of coordinate numbers (for a plane) / a triple of coordinates (for the space).

But this thing we can simply continue in the algebraic way:

For any positive integer $n$, we can define a (canonical) $n$ dimensional vector space: $\Bbb R^n$ consists of the $n$-tuples of real numbers. You can add them and multiply by any real number, coordinatewise. You can check the conditions that it indeed defines a vector space in the abstract sense.

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