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How to visualise bilinear form and inner products?

  1. Dec 12, 2013 #1
    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 havent managed to figure out how to visualise this yet. Could someone please explain it to me in a visual way?
    I cant understand mathematics when I can't visualise it.
  2. jcsd
  3. Dec 12, 2013 #2


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    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 [itex] \left< u, v \right> = \cos(\theta) [/itex] 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.
  4. Dec 12, 2013 #3


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    For completeness, the norm with respect to which [itex]u[/itex] and [itex]v[/itex] are unit vectors is
    [tex]\|u\| = (\langle u, u \rangle)^{1/2}[/tex]
  5. Dec 12, 2013 #4
    The less detailed version of the correct explanations above:

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