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3.7.4 Operations on Scalars, Vectors and Matrices
Most mathematical functions in Mathematica are set up to apply themselves separately to each element in a list. This is true in particular of all functions that carry the attribute Listable.
A consequence is that most mathematical functions are applied element by element to matrices and vectors.
The Log applies itself separately to each element in the vector.
In[1]:= Log[ {a, b, c} ]
Out[1]= 
The same is true for a matrix, or, for that matter, for any nested list.
In[2]:= Log[ {{a, b}, {c, d}} ]
Out[2]= 
The differentiation function D also applies separately to each element in a list.
In[3]:= D[ {x, x^2, x^3}, x ]
Out[3]= 
The sum of two vectors is carried out element by element.
In[4]:= {a, b} + {ap, bp}
Out[4]= 
If you try to add two vectors with different lengths, you get an error.
In[5]:= {a, b, c} + {ap, bp}
Thread::tdlen: Objects of unequal length in {a, b, c} + {ap, bp} cannot be combined.
Out[5]= 
This adds the scalar 1 to each element of the vector.
In[6]:= 1 + {a, b}
Out[6]= 
Any object that is not manifestly a list is treated as a scalar. Here c is treated as a scalar, and added separately to each element in the vector.
In[7]:= {a, b} + c
Out[7]= 
This multiplies each element in the vector by the scalar k.
In[8]:= k {a, b}
Out[8]= 
It is important to realize that Mathematica treats an object as a vector in a particular operation only if the object is explicitly a list at the time when the operation is done. If the object is not explicitly a list, Mathematica always treats it as a scalar. This means that you can get different results, depending on whether you assign a particular object to be a list before or after you do a particular operation.
The object p is treated as a scalar, and added separately to each element in the vector.
In[9]:= {a, b} + p
Out[9]= 
This is what happens if you now replace p by the list {c,d}.
In[10]:= % /. p -> {c, d}
Out[10]= 
You would have got a different result if you had replaced p by {c,d} before you did the first operation.
In[11]:= {a, b} + {c, d}
Out[11]= 
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