25 Sets and Linear Function Spaces
This appendix is taken from Professor Panayiotis Papadopoulos’s notes for UC Berkeley’s ME280A course, Section 2.1.
25.1 Sets
A set \(\mathcal{U}\) is a collection of elements. Elements of the set are denoted \(u \in \mathcal{U}\). A set can be finite or infinite, and may have algebraic structure imposed on it.
25.2 Linear (Vector) Spaces
A set \(\mathcal{V}\) is a linear space (or vector space) over the real numbers \(\mathbb{R}\) if it is equipped with two operations:
- Addition: \({\bf u}+{\bf v} \in \mathcal{V}\) for all \({\bf u},{\bf v} \in \mathcal{V}\),
- Scalar multiplication: \(\alpha{\bf u} \in \mathcal{V}\) for all \(\alpha \in \mathbb{R}\), \({\bf u} \in \mathcal{V}\),
satisfying the standard axioms (commutativity, associativity, existence of zero element, existence of additive inverse, distributivity).
Examples of linear spaces: \(\mathbb{R}^n\); the space of continuous functions on an interval; the space of \(n\times n\) matrices.
25.3 Linear Transformations
A map \({\bf T}: \mathcal{V} \to \mathcal{W}\) between two linear spaces is a linear transformation if: \[\begin{align} {\bf T}({\bf u}+{\bf v}) &= {\bf T}{\bf u}+{\bf T}{\bf v} \quad \text{for all } {\bf u},{\bf v} \in \mathcal{V},\\ {\bf T}(\alpha{\bf u}) &= \alpha{\bf T}{\bf u} \quad \text{for all } \alpha \in \mathbb{R},\; {\bf u} \in \mathcal{V}. \end{align}\]
When \(\mathcal{V} = \mathcal{W}\) is the space of 3D vectors \(\mathbb{E}^3\), a linear transformation from \(\mathbb{E}^3\) to \(\mathbb{E}^3\) is called a tensor.
The rotation \({\bf Q}\) is a linear transformation from \(\mathbb{E}^3\) to \(\mathbb{E}^3\), so it is a tensor. This justifies calling it the rotation tensor throughout the kinematics of rigid bodies chapter.
25.4 Basis and Dimension
A set \(\{{\bf e}_1,\ldots,{\bf e}_n\}\) is a basis for \(\mathcal{V}\) if: - The vectors are linearly independent: \(\sum_i \alpha_i{\bf e}_i = {\bf 0}\) implies \(\alpha_i = 0\) for all \(i\). - They span \(\mathcal{V}\): every \({\bf v} \in \mathcal{V}\) can be written as \({\bf v} = \sum_i v_i{\bf e}_i\).
The number of basis vectors is the dimension of the space.
For 3D Euclidean space \(\mathbb{E}^3\), any orthonormal set \(\{{\bf E}_1,{\bf E}_2,{\bf E}_3\}\) with \({\bf E}_i\cdot{\bf E}_j = \delta_{ij}\) is a basis.