I am self learning Linear Algebra using "Linear Algebra Done Right" a book by Sheldon Axler. And for me its important to understand the smallest thing mentioned in the book. Can someone explain to me what the author means by OVER R/C?
And there should be a reason he mentioned this (an importance), can someone explain me that? This is what the book says :
$\mathbb{R^n}$ is a vector space over $\mathbb{R}$,
and $\mathbb{C^n}$ is a vector space over $\mathbb{C}$
Best Answer
One of the ingredients of a vector space is a definition of scalar multiplication; but you need to know what field the scalars are in! A vector space $V$ over $\mathbb{F}$ has as part of its data a map (scalar multiplication) $\mathbb{F}\times V\to V$. The same set $V$ could be given a different vector space structure with a multiplication map $\mathbb{K}\times V$; this is most common when $\mathbb{K}\subset\mathbb{F}$, and the second map is simply a restriction of the first one.
For example, $\mathbb{C}$ is a vector space over $\mathbb{C}$ as you mention in the question, but it is also a vector space over $\mathbb{R}$, since you can simply restrict scalar multiplication to real scalars. These two structures are quite different; for example, the single element $1$ is a basis of $\mathbb{C}$ over $\mathbb{C}$, since every complex number is of the form $z\cdot 1$ for some unique $z\in\mathbb{C}$. Over $\mathbb{R}$, one possible basis is $1,i$, since every complex number is of the form $a+bi$ for a unique pair $(a,b)\in\mathbb{R}^2$. Notice that the dimension of $\mathbb{C}$ changed depending on whether we gave it the structure of a vector space over $\mathbb{C}$ or over $\mathbb{R}$.