What is meant by compex dimension? (Abstract algebra)

[Ineedhelpimbadatphys]

Homework Statement

Show that the set of n:th order complex polynomials
Pn ≡{a0 +a1z+a2z2 +···+anzn|a0,a1,...,an ∈Cn}
is a vector space. What is its (complex) dimension?

Relevant Equations

Pn ≡{a0 +a1z+a2z2 +···+anzn|a0,a1,...,an ∈Cn}

picture since the text is a little hard to read

i have no problem showing this is a vector space, but what is meant by complex dimention?
Is it just the number on independant complex numbers, so n?

 

[fresh_42]

Do you know how the dimension of a vector space is defined? It involves linear (in-)dependency and therefore the underlying scalar field. E.g. ##\mathbb{C}\cdot [\vec{1}]## is a complex line, hence one-dimensional. But if we consider ##[\vec{1}]## as a real vector, what is the dimension of ##\mathbb{R}\cdot [\vec{1}]##?

 

[Ineedhelpimbadatphys]

There are two independant scalars in a complex number. So does that mean 2n.

 

[Ineedhelpimbadatphys]

Do you know how the dimension of a vector space is defined? It involves linear (in-)dependency and therefore the underlying scalar field. E.g. ##\mathbb{C}\cdot [\vec{1}]## is a complex line, hence one-dimensional. But if we consider ##[\vec{1}]## as a real vector, what is the dimension of ##\mathbb{R}\cdot [\vec{1}]##?

There are two independant scalars in a complex number. So does that mean 2n.

Sorry, this was supposed to be a reply. Im really not understanding the subject so sorry for simple questions.

 

[fresh_42]

There are two independant scalars in a complex number. So does that mean 2n.

No. It only means ##2n## over the reals! It is still ##n## over the complex numbers.

If we write a complex number ##a+\mathrm{i} b## as real vector ##(a,b)## then we get
$$
\dim_\mathbb{R} \left\{\mathbb{R}\cdot\begin{pmatrix}a\\0\end{pmatrix}\oplus \mathbb{R}\cdot\begin{pmatrix}0\\b\end{pmatrix}\right\}=2\, , \,\dim_\mathbb{R} \mathbb{R}\cdot\begin{pmatrix}a\\b\end{pmatrix}=1
$$
and
$$
\dim_\mathbb{C} \left\{\mathbb{C}\cdot a + \mathbb{C}\cdot \mathrm{i}b\right\}=1\, , \,\dim_\mathbb{C} \mathbb{C}\cdot (a+\mathrm{i}b) =1
$$
I assume the exercise was to understand this difference. The field is essential here.

 

[martinbn]

The dimension is ##n+1##.