Naked Science Forum

Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Nimmie on 05/09/2017 19:03:56

Title: what is difference between basis of a vector and components of a vector?
Post by: Nimmie on 05/09/2017 19:03:56

What is the difference between basis of a vector, components of a vector? Also what is a vector space? Please explain like you would do to a six year old boy.

Title: Re: what is difference between basis of a vector and components of a vector?
Post by: jeffreyH on 05/09/2017 20:53:27

Basis vectors are the set of linearly independent vectors of a vector space that can be combined to produce any other vector in the space.
https://en.m.wikipedia.org/wiki/Basis_(linear_algebra) (https://en.m.wikipedia.org/wiki/Basis_(linear_algebra))
It would be instructive for you to study linear algebra. The book Linear Algebra for Dummies is a good first start.

Title: Re: what is difference between basis of a vector and components of a vector?
Post by: evan_au on 06/09/2017 07:00:29

Quote from: Nimmie

what is difference between basis of a vector and components of a vector?

If you have studied Cartesian coordinates in high school, I can explain it in 2 dimensions:
- Pick a random point A on the graph. In Cartesian coordinates, we label A as (x,y).
- Draw a line from the Origin (0,0) to A (x,y). This is your vector.
- The way we measure the location of point A and the Origin is via the X and Y axes. These are the basis vectors for the Cartesian space.

We can define that in 3 dimensions by adding a new basis vector, the Z axis.
- 4 or more dimensions play games with your mind
- But you can imagine it like the flight of an aeroplane - it has a physical location (3 dimensions) plus a linear velocity (3 more dimensions) plus a couple more if you want to control roll, yaw and pitch.

Quote

Please explain like you would do to a six year old

Sorry - I could only get it down to the level of a 16 year old playing Flight Simulator.

Title: Re: what is difference between basis of a vector and components of a vector?
Post by: jeffreyH on 06/09/2017 12:49:32

If talking of n dimensional basis vectors you come up against Hilbert Spaces.
https://en.m.wikipedia.org/wiki/Hilbert_basis (https://en.m.wikipedia.org/wiki/Hilbert_basis)