How to multiply two vector and get a matrix?

Normal matrix multiplication works as long as the vectors have the right shape. Remember that * in Numpy is elementwise multiplication, and matrix multiplication is available with numpy.dot() (or with the @ operator, in Python 3.5)

>>> numpy.dot(numpy.array([[1], [2]]), numpy.array([[3, 4]]))
array([[3, 4],
       [6, 8]])

This is called an “outer product.” You can get it using plain vectors using numpy.outer():

>>> numpy.outer(numpy.array([1, 2]), numpy.array([3, 4]))
array([[3, 4],
       [6, 8]])