Notice that these axioms have a lot of overlap with the field axioms. Because of that, many of the same results (like "additive inverses are unique") hold in a vector space, using exactly the same proof.
The set \(\mathcal{F}\) of all functions \(\mathbb{R}\to\mathbb{R}\) is a vector space with coefficients in \(\mathbb{R}\text{,}\) if we use the operations
Properly speaking, the study of vector spaces belongs in a linear algebra class. We won’t go into serious detail here because we’re about other business. But you should learn linear algebra, and a lot of it.
Given two vector spaces (call them \(V\) and \(W\)), we can ask which functions \(F:V\to W\) are mutually compatible with the operations on \(V\) and \(W\text{.}\) That is, we want \(F\) to satisfy:
The words function, transformation, and map will be used synonymously in these notes, but with a stylistic preference: I will habitually write linear transformation or linear map, while keeping the word function for any assignment of outputs to inputs.
If we take \(V=\mathbb{R}\) and \(W=\mathbb{R}\text{,}\) equipped with the ordinary operations you know and love. Suppose \(F:\mathbb{R}\to\mathbb{R}\) is a linear transformation. Define \(a=F(1)\text{.}\)
A function \(F:\mathbb{R}^n\to\mathbb{R}^m\) is linear if and only if it comes from matrix multiplication. That is, a function \(F(x_1,\ldots,x_n)=(y_1,\ldots,y_m)\) is linear if and only if there is some \(m\times n\) matrix \(A\) so that
It’s not quite true that the theory of linear transformations is the same as the study of matrices, but it’s close enough to being true for our purposes.
If you look back up at the ideas we used in Chapter 1 and Chapter 2, the absolute value function plays a big role. Informally, \(\lvert r\rvert\) tells us how big \(r\in\mathbb{R}\) is. To define convergence of limits, we need to ask how big \(x_n-L\) is. At the very least, we’ll want to have that same kind of functionality.
The closed unit ball in \(\mathbb{R}^2\text{,}\) equipped with the norm \(\lVert\cdot\rVert_p\text{.}\) Use the slider to change the value of \(p\) and see the effect on the unit ball.
\begin{equation*}
\mathbb{R}^\infty=\left\{ \left(x_1,x_2,\ldots,x_k,\ldots\right)\middle\vert x_i\in \mathbb{R}\text{ and only finitely many }x_i\neq 0\right\}
\end{equation*}
This is a vector space with componentwise addition and scaling.
The notation \(\lVert\cdot\rVert\) is reminiscent of absolute value, and this is a conscious choice. It allows us to transfer most notions from an ordered field into a normed space. (But not, of course, the order part or the multiplicative inversion part -- then we’d just land back in an ordered field.)
