Bilinear Form Confusion:
So, bilinear forms use to be confusing to me because I didn’t know what either bilinear meant or what forms meant. But, a form is, very often, just a function. Map is much more common among mathematicians but they’re just functions. They take in inputs and give you outputs. The same functions we’ve been using since middle school.
Now, they take in two vectors and give you back a number. That’s it. The most famous example is the dot product, which even high schoolers know. You take in 2 vectors and you output a number. That’s it.
Bilinear forms are a function that take in 2 vectors and give you back a number.
They have the word “linear” in them because if you plug in (a+b,c), it’s the same as plugging in (a,c) + (b,c). Same for (a,b+c).
If you have a symmetric bilinear form it means that (a,b) = (b,a). Skew-symmetric means what you think antisymmetric means (a,b) = -(b,a). So, like cross products.
Next, I’ll be using bilinear forms to define the symplectic group.
