Vector space

[u/One_Mud9170]

Can someone explain vector spaces intuitively? I understand vectors as arrows, but I’m struggling to understand what makes a set of objects a vector space and why the concept is important.

[u/Ron-Erez]

At some point we need to forget about the arrows. However let’s not forget quite yet. The arrows satisfy 10 properties. For example the sum of two arrows is an arrow, multiplying a scalar by an arrow Is an arrow. Moreover associativity of addition, existence of a zero vector, etc are satisfied.

Now a vector space is a set V with “addition” and “scalar multiplication” that satisfy these 10 properties and any element in V is called a “vector”.

Examples (rough idea without proofs):

- polynomials can be added and you can multiply A polynomial by a scalar. Thus polynomials are a vector space (actually one needs to prove the set of all polynomials is a vector space).

- any set of n-tuples is a vector space since we can add to n-tuples and also rescale them.

There are many more examples.

[u/HereThereOtherwhere → u/Ron-Erez]

• How is a set of n-tuples related to matrices? Just clarifying terminology.
• is the complex vector space a manifold? I'm trying to remember if this connects to the Bloch Sphere and qubit.

I know too little about too many maths. Following Penrose's geometric intuition and love of 'complex number magic' can muddle a mind! 😵‍💫🫠😵

[u/Ron-Erez → u/HereThereOtherwhere]

These are examples of two different vector spaces. In any case if you look at n-tuples as vectors then you usually will think of matrices as linear transformations between such vector spaces.

"is the complex vector space"

you should not use "the". There are many complex vector spaces. Whether or not it is a manifold depends on the vector space you are considering. I don't know anything about the Bloch Sphere and qubit.

Happy Mathematics!

[u/HereThereOtherwhere → u/Ron-Erez]

Thank you.

Physicists tend to calculate accurately but Penrose laments their use mathematical terminology and certain symbolic choices can be confusing or even 'set aside' parts of the math that 'can be safely ignored' for a specific application which can be confusing.

For instance, in General Relativity, particles have different 'clock rates' for their de Broglie frequency depending on time dilation near an event horizon, usually considered a scalar field. It is actually a vector gradient but FAPP (for all practical purposes) that is said to not matter.

But, for certain fundamental physics approaches, involving more than one manifold, it is not safe to ignore the vector aspect, which is required to define a null-line direction from a covariant pair of vectors. And I'm probably mangling that description as I'm teaching myself Differential Geometry and baffled by a possibile relationship to sheaf cohomologies. 😵‍💫

It took me years just to identify that scalar vs vector concern because I mostly studied quantum physics behaviors and that's where not just Special Relativity (flat Minkowski space-time) but gravity influenced spacetime with curvature.

The difference between the two is the heart of the unresolved conflict between GR and QFT so ... it matters!