Tensor product

Okay kiddo, imagine you have two sets of blocks – one set has red blocks and the other set has blue blocks. Now, you want to make a new set of blocks which has all possible combinations of these red and blue blocks (putting a red block next to a blue block, putting two red blocks together or two blue blocks together, etc). That new set of blocks that you made is called the tensor product.

In math, we use a special symbol to represent the tensor product, which looks like a letter ‘X’ with a hat on it (or it can also look like the direct product symbol ‘⊗’ which is like a sideways T with a little circle). It tells us that we're multiplying two things together to create a new thing.

But in math, we don't use blocks, we use something called 'vectors', which are like arrows. Each vector has a starting point (which is usually at the origin or center) and a direction (which tells us which way the arrow points). Just like with blocks, we can take two vectors and create a new vector by combining them, and this is where the tensor product comes in.

When we take the tensor product of two vectors, we end up with a special type of vector called a tensor, which has properties that make it different from a regular vector. But basically, it's just a way for us to combine two vectors into a new one that has all possible combinations of their directions.

For example, if we had a red vector that points in the direction of 'x', and a blue vector that points in the direction of 'y', then the tensor product of those two vectors would be a new tensor that contains four vectors: one that points in the direction of 'x', one that points in the direction of 'y', and two more that point diagonally in between 'x' and 'y'.

Overall, the tensor product is a fancy way for us to multiply vectors together in a way that lets us create new vectors with lots of different combinations of properties. It's sort of like playing with blocks, but for grown-up math!