ℝⁿ
\(\mathbb{R}^n\)
Let \(\mathbb{N}\) be the nonnegative integers and let \(n \in \mathbb{N}\).
Define \([n]=\{i \in \mathbb{N}: i < n\}\). In particular, \([0]=\emptyset\).
Let \(\mathbb{R}^n=\mathbb{R}^{[n]}\), the set of functions \([n] \to \mathbb{R}\).
\(n=0\)
\(\mathbb{R}^0 = \mathbb{R}^\emptyset = \{\emptyset\}\). 1
\(\mathbb{R}^0\) is an \(\mathbb{R}\)-linear space, with one element, ∅.
\(n \geq 1\)
Let \(n \in \mathbb{N}_{\geq 1}\). For \(x,y \in \mathbb{R}^n\), we define \(x+y \in \mathbb{R}^n\) by \((x+y)(i)=x(i)+y(i)\), \(i \in [n]\).
For \(x \mathbb{R}^n\) and \(c \in \mathbb{R}\), we define \(cx \in \mathbb{R}^n\) by \((cx)(i)=cx(i)\), \(i \in [n]\). Then \(\mathbb{R}^n\) is an \(\mathbb{R}\)-linear space.
For \(i,j \in \mathbb{N}\), define
\[\delta_{i,j} = \begin{cases}1&i = j\\ 0&i \neq j \end{cases}\]For \(k \in [n]\) define \(e^k \in \mathbb{R}^n\) by
\[e^k(i) = \delta_{i,k}, \qquad i \in [n].\]\(\{e^k: k \in [n]\}\) is a basis for \(\mathbb{R}^n\).
Dual space \((\mathbb{R}^n)^*\)
Let \((\mathbb{R}^n)^*\) be the set of linear maps \(\mathbb{R}^n \to \mathbb{R}\), the dual space of \(\mathbb{R}^n\).
For \(x \in \mathbb{R}^n\), define \(x^T \in (\mathbb{R}^n)^*\) by
\[x^T y = \sum_{i \in [n]} x(i)y(i),\qquad y \in \mathbb{R}^n.\]We call \(x^T\) the transpose of \(x\): \(x\) is a vector/column vector and \(x^T\) is a covector/row vector.
\(\{(e^k)^T: k \in [n]\}\) is a basis for \((\mathbb{R}^n)^*\).
Write $Tx=x^T\(. The map\)T:x \mapsto x^T\(is a linear isomorphism\)\mathbb{R}^n \to (\mathbb{R}^n)^*$$.
Basis expansion
For \(x \in \mathbb{R}^n\) and for \(k \in [n]\), we have
\[x^T e^k = \sum_{i \in [n]} x(i) e^k(i) = \sum_{i \in [n]} x(i) \delta_{i,k} = x(k)\]and
\((e^k)^T x = \sum_{i \in [n]} e^k(i) x(i) = \sum_{i \in [n]} \delta_{i,k} x(i)= x(k)\).
Using this, one then works out that
\[x = \sum_{k \in [n]} ((e^k)^T x)e^k.\]Linear algebra
For real finite dimensional vector spaces \(V\) and \(W\), let \(\mathscr{L}(V,W)\) be the set of linear transformations \(V \to W\), which is itself a real finite dimensional vector space.
\[(\mathbb{R}^n)^* = \mathscr{L}(\mathbb{R}^n,\mathbb{R}).\]An \(m \times n\) matrix is an element of \(\mathscr{L}(\mathbb{R}^n,\mathbb{R}^m)\) and a choice of basis for \(\mathbb{R}^n\).
\[\dim \mathscr{L}(V,W) = \dim V \cdot \dim W.\]In particular,
\[\dim \mathscr{L}(\mathbb{R}^m,\mathbb{R}^n) = \dim \mathbb{R}^m \cdot \dim \mathbb{R}^n = m\cdot n.\]Transposes of linear maps
Let \(A \in \mathscr{L}(\mathbb{R}^n,\mathbb{R}^m)\).
Define \(A^T \in \mathscr{L}((\mathbb{R}^m)^*,(\mathbb{R}^n)^*)\) by
\[A^T f x = f(Ax),\qquad f \in (\mathbb{R}^m)^*, x \in \mathbb{R}^n,\]called the transpose of \(A\).
Transposes of matrices
We remind ourselves that \(\{(e^k)^T: k \in [m]\}\) is a basis for \((\mathbb{R}^m)^{*}\) and \(\{e^j^T: j \in [n]\}\) is a basis for \(\mathbb{R}^n\). Thus, \(A^T \in \mathscr{L}((\mathbb{R}^m)^*,(\mathbb{R}^n)^*)\) is also defined by
\[A^T (e^k)^T e^j = (e^k)^T(Ae^j),\qquad j \in [n], k \in [m].\]