Well, a bilinear map is just a function that takes two vectors as input and spits out another vector. That’s it! But there’s a catch: this function has to be linear in both of its arguments. Let me explain what I mean by that.

Let’s say we have some fancy math space called V, where all the cool kids hang out. And let’s say we want to define a bilinear map between two vectors in V call them v and w. So we write: f(v,w) = something.

Now, for this function to be linear in its first argument (let’s call it x), that means if we add some other vector y to x, the output of our bilinear map should also change by adding the result of applying the same function with y instead of x: f(x+y,w) = f(x,w) + f(y,w).

And similarly for linearity in its second argument (let’s call it z): if we add some other vector u to z, then our bilinear map should also change by adding the result of applying the same function with u instead of z: f(v,z+u) = f(v,z) + f(v,u).

So that’s what makes a bilinear map so special it has to be linear in both arguments. But why do we care about these things? Well, turns out they come up all the time in math and physics! For example, if you have two vectors representing forces acting on an object (let’s call them F1 and F2), then their combined effect is just a new vector that represents the total force. This new vector can be found using a bilinear map specifically, we add the components of each individual force together to get our final result: f(F1,F2) = (F1_x + F2_x, F1_y + F2_y, F1_z + F2_z).

Now tensor products. A tensor product is a way of combining two vector spaces into one bigger space that allows us to do all sorts of cool math stuff. The idea behind it is pretty simple: we take every possible pairing of vectors from our original spaces, and then stick them together in some fancy way to create new “tensor” objects.

For example, let’s say we have two vector spaces V and W (just like before). We can define a tensor product space called VW that contains all possible pairs of vectors from V and W: vw for every vV and wW. This new space has dimensions equal to the product of the original spaces, so if V had dimension n and W had dimension m, then our new tensor product space would have dimension nm.

So that’s what bilinear maps and tensor products are all about! They might sound fancy, but they really just involve combining two things in a linear way to get something else. And as we saw with the example of forces acting on an object, these concepts can be incredibly useful in real-world applications.

So next time you’re feeling overwhelmed by some math problem or physics concept, remember: sometimes all it takes is breaking things down into their simplest parts and combining them linearly!