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\$$.