Section 23.2 Terminology and notation
- \(n\)-dimensional complex vector
a vector with \(n\) complex components; i.e. a vector \(\uvec{v} = (v_1,v_2,\dotsc,v_n)\) where each component \(v_j\) is a complex number
- complex scalar multiplication
a rule for associating to a complex number \(k\) and an object \(\uvec{v}\) another object \(k\uvec{v}\)
- complex vector space
a collection of mathematical objects, along with appropriate conceptions of vector addition and complex scalar multiplication, that satisfies the Vector space axioms (where we interpret scalar to mean complex number instead of just real number)
Here follows the notation we will use for some common complex vector space examples.
- \(\C^n\)
the collection of all \(n\)-dimensional complex vectors
- \(\matrixring_{m \times n}(\C)\)
the vector space of all \(m\times n\) matrices with entries that are complex numbers; as in the real case before, when \(m=n\) we sometimes just write \(\matrixring_n(\C)\) to mean the vector space of all square \(n\times n\) complex matrices
- \(\poly(\C)\)
the vector space of all polynomials with complex coefficients in a single complex variable
- \(\poly_n(\C)\)
the vector space of all polynomials with complex coefficients in a single complex variable that have degree \(n\) or less
Finally, we will see in Subsection 23.3.4 that every complex vector space can also be considered as a real vector space. So in some of our notation we will occasionally need to make it clear which point of view is being used.
- \(\Span_{\R} S\)
the collection of all possible linear combinations of the vectors in set \(S\text{,}\) where only real scalars are allowed as the coefficients
- \(\Span_{\C} S\)
the collection of all possible linear combinations of the vectors in set \(S\text{,}\) where complex scalars are allowed as the coefficients
- \(\dim_{\R} V\)
the dimension of the real vector space \(V\)
- \(\dim_{\C} V\)
the dimension of the complex vector space \(V\)