What is the difference between a tensor, vector, and a matrix?

Question

I'm currently going through notes on a physics course and I'm having trouble understanding the difference between a tensor, a vector, and a matrix. I know that a vector is a kind of tensor and that a matrix is also a form of tensor, but what I'm having trouble understanding is how they differ, and how a vector is related to a matrix.

Answer 1

Short and a little inaccurate answer: vector is one-dimensional tensor, matrix is a two-dimensional tensor.

More details now:

Tensors are multidimensional arrays which have certain properties. Not every multidimensional array is a tensor, check this discussion for more details.

There are two types of one-dimensional tensors: vectors and co-vectors. Both vectors and co-vectors can be represented as a simple array of numbers. The difference between those two come out when you have the array of numbers which represent the object in one basis and want to find out what numbers represent the same object in some other basis. Transformation rules are slightly different for vectors and co-vectors. Vectors and co-vectors are usually represented as "column of numbers" and "line of numbers" respectively.

So, vector is always a one-dimensional tensor, if you have a one-dimensional tensor it is either a vector or co-vector.

Two-dimensional tensors are called matrices. There are not two but four different types of two-dimensional tensors, but there are no special names for them. As in case with vectors transformation rules are slightly different when you go from one basis to another, but there are no special names for these tensors: all of them are just matrices.

Actually sometimes they call any two-dimensional array a "matrix". Even if it is not a tensor at all. Again, for more details about the difference between array and tensor refer to the discussion I mentioned earlier.