Exploring Inner Products in Vector Spaces: The Impact on Geometry

[talolard]

Homework Statement

Hey, I ahve a curioisity (not homework )question.
We learned that there are an infinite number of inner products that can be defined ona vector space and that inner product space is what gives us the notion of distance within a given space.
So if we defined some non standard inner product on R^n what would that mean in terms of geoemtry?
For example, say I looked at R^2 with the standard inner product and defined a square with vectors. Then I "took a new space" in R^2 but with some other inner product and plotted the same vectors, what would they still be a square?


Thanks
Tal

Homework Equations

The Attempt at a Solution

 

[radou]

Try to define a new norm on R^2 and see if it preserves orthogonality, for example.

 

[HallsofIvy]

Homework Statement

Hey, I ahve a curioisity (not homework )question.
We learned that there are an infinite number of inner products that can be defined ona vector space and that inner product space is what gives us the notion of distance within a given space.
So if we defined some non standard inner product on R^n what would that mean in terms of geoemtry?
For example, say I looked at R^2 with the standard inner product and defined a square with vectors. Then I "took a new space" in R^2 but with some other inner product and plotted the same vectors, what would they still be a square?


Thanks
Tal

Homework Equations

The Attempt at a Solution

Changing the inner product is essentially the same as changing your basis which, in $R^n$ is the same as changing your coordinate system. In particular, if you chose <(x,y),(u,v)> = xu+ 2yv as your inner product, an orthonormal basis would be $\{(1, 0), (0, \sqrt{2}/2)\}$ so your vertical axis would be squashed compared to your horizontal axis. No, it would not still be a square.

 

[talolard]

Cool.
Thank you.