You also have to understand that a^T b is a popular way of writing the dot product. <a,b> is for dumb high schoolers. Then there is the fun part in German that

You also have to understand that a^T b is a popular way of writing the dot product. <a,b> is for dumb high schoolers. Then there is the fun part in German that the dot product and inner product are both called Skalarprodukt. There are many inner products of which the dot product is only one.