Keith wrote:

> I've never been in a Linear algebra, how does one calculate the dual of a vector space? And what is it used for?

Let me take the second question first. In a basic math class you'd call

\[ f(x) = a x + b \]

a linear function, but in advanced math we call this **affine**, and say a typical linear function of one variable is like

\[ f(x) = a x .\]

Linear functions are the simplest functions except for constant functions.

For any vector spaces \\(V\\) and \\(W\\) over a field \\(k\\) we say a map \\(f \colon V \to W \\) is **linear** iff \\(f((cv+dw) = c f(v) + d f(w) \\) for all \\(c,d \in k\\) and \\(v,w \in V\\). We call the set of linear maps \\(f \colon V \to k\) the **dual** of \\(V\\) and it turns out to be a vector space in its own right, which we call \\(V^\ast\\). It's the starting point for all studies of functions on \\(V\\). For example, polynomial function \\(f \colon V \to k\\) are linear combinations of products of linear functions.

Here's a nice fact about the dual: the set of all linear maps from a vector space \\(V\\) to a vector space \\(W\\) is itself a vector space, and this vector space is isomorphic to \\(W \otimes V^\ast\\).

> I've never been in a Linear algebra, how does one calculate the dual of a vector space? And what is it used for?

Let me take the second question first. In a basic math class you'd call

\[ f(x) = a x + b \]

a linear function, but in advanced math we call this **affine**, and say a typical linear function of one variable is like

\[ f(x) = a x .\]

Linear functions are the simplest functions except for constant functions.

For any vector spaces \\(V\\) and \\(W\\) over a field \\(k\\) we say a map \\(f \colon V \to W \\) is **linear** iff \\(f((cv+dw) = c f(v) + d f(w) \\) for all \\(c,d \in k\\) and \\(v,w \in V\\). We call the set of linear maps \\(f \colon V \to k\) the **dual** of \\(V\\) and it turns out to be a vector space in its own right, which we call \\(V^\ast\\). It's the starting point for all studies of functions on \\(V\\). For example, polynomial function \\(f \colon V \to k\\) are linear combinations of products of linear functions.

Here's a nice fact about the dual: the set of all linear maps from a vector space \\(V\\) to a vector space \\(W\\) is itself a vector space, and this vector space is isomorphic to \\(W \otimes V^\ast\\).