From "Linear algebra done right" by S. Axler:
"a vector space over $\mathbb R $ is called a real vector space and a vector space over $\mathbb C $ is called a complex vector space"
Does this imply that the Field (by which I mean the type of the scalar used for multiplication) is systematically also the type of the coordinates of the vector ?
Do we ever study the case where the scalar used for multiplication is in $\mathbb R $ but the vector coordinates are in $\mathbb C $ , or vice – versa ?
Best Answer
We speak of a finite-dimensional vector space $V$ over a given field $\mathbf F$. If $\mathcal B=(u_1,\dotsc,u_k)$ is a basis for $V$, then any vector $v\in V$ may be written as a linear combination of the basis vectors: $$ v = c_1u_1+\dotsb+c_ku_k $$ where $c_1,\dotsc,c_k$ are the coordinates of $v$ with respect to $\mathcal B$. The coordinates $c_1,\dotsc,c_k$ each live inside of the field $\mathbf F$.