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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} ]


The same is true for a matrix, or, for that matter, for any nested list.

In[2]:= Log[ {{a, b}, {c, d}} ]


The differentiation function D also applies separately to each element in a list.

In[3]:= D[ {x, x^2, x^3}, x ]


The sum of two vectors is carried out element by element.

In[4]:= {a, b} + {ap, bp}


If you try to add two vectors with different lengths, you get an error.

In[5]:= {a, b, c} + {ap, bp}


This adds the scalar 1 to each element of the vector.

In[6]:= 1 + {a, b}


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


This multiplies each element in the vector by the scalar k.

In[8]:= k {a, b}


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


This is what happens if you now replace p by the list {c, d}.

In[10]:= % /. p -> {c, d}


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}