Operations on Scalars, Vectors, and Matrices

Most mathematical functions in the Wolfram Language 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.
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The same is true for a matrix, or, for that matter, for any nested list.
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The differentiation function D also applies separately to each element in a list.
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The sum of two vectors is carried out element by element.
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If you try to add two vectors with different lengths, you get an error.
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This adds the scalar 1 to each element of the vector.
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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.
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This multiplies each element in the vector by the scalar k.
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It is important to realize that the Wolfram Language 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, the Wolfram Language 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.
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This is what happens if you now replace p by the list {c,d}.
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You would have gotten a different result if you had replaced p by {c,d} before you did the first operation.
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