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```To all who like overloading, I dearly suggest that they explain to
themselves the following bit of Mathematica:

In[1]:= f[v_] = {{1,0}, {1,1}} . (v + {0,1});

In[2]:= f[{0,0}]

Out[2]= {{0, 0}, {1, 1}}

Firstly, if you do not think about the above, you won't even see what's
wrong. Secondly, once you do see what is wrong, you won't know why it's
wrong. The problem is caused by the meaning of + and Mathematica
evaluation strategy.

Explanation of what is wrong:

1) {0,1} is a two-dimensional vector

2) {{1,0},{1,1}} is the 2x2 matrix with rows (1,0) and (1,1)

3) The operator . means, according to the help system:

In[4]:= ?.
a.b.c or Dot[a, b, c] gives products of vectors, matrices and tensors.

For example, multiplying a 2x2 matrix and a 2D vector gives a 2D
vector:

In[5]:= {{1,0},{1,1}} . {0,1}

Out[5]= {0, 1}

4) Clearly, + is addition :-) but in case you do not trust me:

In[6]:= ?+
x + y + z represents a sum of terms.

5) The definition of f then says: take v, add to it the vector {0,1},
then multiply by the 2x2 matrix {{1,0},{1,1}}.

6) Thus we compute f[{0,0}] by hand according to 5):

f[{0,0}]
= {{1,0},{1,1}} . ({0,0} + {0,1})
= {{1,0},{1,1}} . {0,1}
= {0,1}

7) Hmm, strange, Mathematica thinks the answer is a 2x2 matrix, and we
think it's a 2D vector.