0
$\begingroup$

My lecture notes say that: ''Given any bilinear form $\tau$ on $V$, there is a uniquely determined linear operator $T$ on $V$ such that $$\tau(v,w) = v \, .(Tw)$$ so once we've fixed a 'starting' bilinear form (the inner product), we can get any other bilinear form $\tau$ from this via a linear operator''.

Why is it necessary to fix a 'starting' point and what does this 'starting' point actually mean? Also is there a way of proving the above?

  • 0
    can you explain the dot notation? usually one thinks of bilinear forms as homomorphisms $V\rightarrow V^*$ and not "operators"2012-04-02
  • 1
    What it means is: start with your favorite bilinear form for $V$. Then you can "translate" any other bilinear form you may encounter lying on the street into your favorite one through the use of a linear operator. You don't need to "start" by fixing one, but the point is that you can agree to have a "prefered" bilinear form, and then translate any other bilinear form into that one.2012-04-02
  • 2
    Your starting bilinear form can't be arbitrary (e.g., it can't be 0). The meaning of the starting point is to fix a way of thinking about $V$ as a concrete vector space $F^n$. For a proof of what you ask about, see Section 2 of http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/bilinearform.pdf2012-04-02

1 Answers 1