When I tried to compute w^T * x using numpy, it was super confusing for me as well. In fact, I couldn't implement it myself. So, this is one of the few gotchas in NumPy that we need to acquaint ourselves with.
As far as 1D array is concerned, there is no distinction between a row vector and column vector. They are exactly the same.
Look at the following examples, where we get the same result in all cases, which is not true in [the theoretical sense of] linear algebra:
In [37]: w
Out[37]: array[[0, 1, 2, 3, 4]]
In [38]: x
Out[38]: array[[1, 2, 3, 4, 5]]
In [39]: np.dot[w, x]
Out[39]: 40
In [40]: np.dot[w.transpose[], x]
Out[40]: 40
In [41]: np.dot[w.transpose[], x.transpose[]]
Out[41]: 40
In [42]: np.dot[w, x.transpose[]]
Out[42]: 40
With that information, now let's try to compute the squared length of the vector |w|^2.
For this, we need to transform w to 2D array.
In [51]: wt = w[:, np.newaxis]
In [52]: wt
Out[52]:
array[[[0],
[1],
[2],
[3],
[4]]]
Now, let's compute the squared length [or squared magnitude] of the vector w :
In [53]: np.dot[w, wt]
Out[53]: array[[30]]
Note that we used w, wt instead of wt, w [like in theoretical linear algebra] because of shape mismatch with the use of np.dot[wt, w]. So, we
have the squared length of the vector as [30]. Maybe this is one of the ways to distinguish [numpy's interpretation of] row and column vector?
And finally, did I mention that I figured out the way to implement w^T * x ? Yes, I did :
In [58]: wt
Out[58]:
array[[[0],
[1],
[2],
[3],
[4]]]
In [59]: x
Out[59]: array[[1, 2, 3, 4, 5]]
In [60]: np.dot[x, wt]
Out[60]: array[[40]]
So, in NumPy, the order of the operands is reversed, as evidenced above, contrary to what we studied in theoretical linear algebra.
P.S. : potential gotchas in numpy
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