How to visualise bilinear form and inner products?

[mathmo94]

Hi I'm taking abstract linear algebra course and having trouble visualising bilinear form and inner products. I can visualise vector spaces, span, dimensions etc but haven't managed to figure out how to visualise this yet. Could someone please explain it to me in a visual way?
I can't understand mathematics when I can't visualise it.

 

[Office_Shredder]

Inner products are just abstracted versions of the dot product,which is really a way of measuring angles and projections. If u and v are unit vectors, then $\left< u, v \right> = \cos(\theta)$ where theta is the angle between u and v. It also equals the length of the projection of u onto v. If they are not unit vectors then you need to scale appropriately.

 

[pasmith]

Inner products are just abstracted versions of the dot product,which is really a way of measuring angles and projections. If u and v are unit vectors, then $\left< u, v \right> = \cos(\theta)$ where theta is the angle between u and v. It also equals the length of the projection of u onto v. If they are not unit vectors then you need to scale appropriately.

For completeness, the norm with respect to which $u$ and $v$ are unit vectors is
$$\|u\| = (\langle u, u \rangle)^{1/2}$$

 

[economicsnerd]

The less detailed version of the correct explanations above:

An inner product yields a notion of length for vectors (where $||x||$ denotes the length of $x$), which agrees quite well with our intuitions about length. For any pair of vectors $x,y$, the number $\langle x,y\rangle$ has size between zero and $||x||\cdot||y||$. If it has size $\approx||x||\cdot||y||$, then $x$ and $y$ are approximately parallel. If it's $\approx 0$, then $x$ and $y$ are approximately perpendicular. If it's a positive number, then $x$ and $y$ make an acute angle (from the origin). If it's a negative number, then $x$ and $y$ make an obtuse angle (from the origin).