Mechanics of Continua and Structures
Definition A vector space is a set of vectors closed under finite vector addition and scaler multiplication.
Definition
$\boldsymbol{a}, \boldsymbol{b} \mapsto \text{Dot}(\boldsymbol{a},\boldsymbol{b}) \in \mathbb{R}$, where $\mathbb{R}$ is the set of real numbers, and $\text{Dot}(\boldsymbol{a},\boldsymbol{b}) = \text{Dot}(\boldsymbol{b},\boldsymbol{a})$.
Definition An inner product on a real vector space $\mathbf{X}$ is a symmetric, bilinear, positive-definite function
\[\begin{align*} \langle \cdot, \cdot \rangle : \mathbf{X} \times \mathbf{X} \rightarrow \mathbb{R} \\ (\textbf{x}^*,\textbf{x}) \mapsto \langle \textbf{x}^*,\textbf{x} \rangle \end{align*}\]Definition A vector space with an inner product defined on it is called an inner product space
Definition A Euclidean space is a real finite-dimensional vector space with an inner product $\langle \cdot, \cdot \rangle$
Definition The Euclidean space $\mathbb{R}$ of real numbers is defined b the inner product \(\begin{align*} \langle x^*,x \rangle := x^*x \end{align*}\)
Definition The Euclidean space $\mathbb{R}^n := \mathbb{R} \times …. \times \mathbb{R}$ (n times), is one in which the elements are n-tuples or ordered-sets of n-real numbers often denoted as $(x_1, x_2,….,x_n)$ or $\left(\boldsymbol{x}_i\right)_{i\in (1,\ldots,n)}$.
The inner product of $(x_i) = x$ and $y = (y_i)$ is defined to be $x_iy_i$.
Let $\mathbf{E}^{n_{\rm sd}}$ be an arbitrary Euclidean space. Let $\boldsymbol{x}_i \in \mathbf{E}^{n_{\rm sd}}$, $i=1,\ldots,m\le n_{\rm sd}$, satisfy the equation
\[\begin{equation} \hat{\boldsymbol{x}}_i\cdot \hat{\boldsymbol{x}}_j=\delta_{ij}. \end{equation}\]Then $\left(\boldsymbol{x}_i\right)_{i\in (1,\ldots,m)}$ is called an orthornormal set of vectors.
Remark An orthornal set of vectors is an independent set of vectors.
Let $\mathbf{E}^{n_{\rm sd}}$ be an arbitrary Euclidean space. We can always find $\hat{\boldsymbol{E}}_i \in \mathbb{E}^{n_{\rm sd}}$, $i=1,\ldots,n_{\rm sd}$, such that $(\hat{\boldsymbol{E}}_i)_{i\in (1,\ldots,n_{sd})}$ is an orthonormal set of vectors.
Since an orthonormal set of vectors are also an independent set of vectors, $(\hat{\boldsymbol{E}}_i)_{i\in (1,\ldots,n_{sd})}$ are $n_{\rm sd}$ independent vectors in $\mathbb{E}^{n_{\rm sd}}$. From which it follows that $(\hat{\boldsymbol{E}}_i)_{i\in (1,\ldots,n_{sd})}$ are $n_{\rm sd}$ provides a set of basis vectors for $\mathbb{E}^{n_{\rm sd}}$. For this reason, $(\hat{\boldsymbol{E}}_i)_{i\in (1,\ldots,n_{sd})}$ is referred to as an orthonormal basis of $\mathbb{E}^{n_{\rm sd}}$.
Specifically, for case of $n_{\rm sd}=3$ we have$
\[\begin{align} \hat{\boldsymbol{E}}_1\cdot\hat{\boldsymbol{E}}_1&=1,\\ \hat{\boldsymbol{E}}_2\cdot\hat{\boldsymbol{E}}_2&=1,\\ \hat{\boldsymbol{E}}_3\cdot\hat{\boldsymbol{E}}_3&=1,\\ \hat{\boldsymbol{E}}_1\cdot\hat{\boldsymbol{E}}_2 &= 0 & \hat{\boldsymbol{E}}_2\cdot\hat{\boldsymbol{E}}_1 &= 0, \\ \hat{\boldsymbol{E}}_2\cdot\hat{\boldsymbol{E}}_3 &= 0 & \hat{\boldsymbol{E}}_3\cdot\hat{\boldsymbol{E}}_2 &= 0, \\ \hat{\boldsymbol{E}}_3\cdot\hat{\boldsymbol{E}}_1 &= 0 & \hat{\boldsymbol{E}}_1\cdot\hat{\boldsymbol{E}}_3 &= 0. \\ \end{align}\]For two arbitrary vectors $\boldsymbol{a}$ and $\boldsymbol{b}$, such that $(\boldsymbol{a})_{\cdot i}=a_i$ and $(\boldsymbol{b})_{\cdot i}=b_i$, we have that
\[\begin{align} (a_i \hat{\boldsymbol{E}}_i) \cdot (b_j \hat{\boldsymbol{E}}_j) = & a_1 b_1 \hat{\boldsymbol{E}}_1 \cdot \hat{\boldsymbol{E}}_1+ a_1 b_2 \hat{\boldsymbol{E}}_1 \cdot \hat{\boldsymbol{E}}_2+ a_1 b_3 \hat{\boldsymbol{E}}_1 \cdot \hat{\boldsymbol{E}}_3+\\ & a_2 b_1 \hat{\boldsymbol{E}}_2 \cdot \hat{\boldsymbol{E}}_1+ a_2 b_2 \hat{\boldsymbol{E}}_2 \cdot \hat{\boldsymbol{E}}_2+ a_2 b_3 \hat{\boldsymbol{E}}_2 \cdot \hat{\boldsymbol{E}}_3+ \\ & a_3 b_1 \hat{\boldsymbol{E}}_3 \cdot \hat{\boldsymbol{E}}_1+ a_3 b_2 \hat{\boldsymbol{E}}_3 \cdot \hat{\boldsymbol{E}}_2+ a_3 b_3 \hat{\boldsymbol{E}}_3 \cdot \hat{\boldsymbol{E}}_3 \\ =& a_1 b_1 + a_2 b_2 + a_3 b_3 \\ =&a_i b_i \end{align}\]Definition Isomorphism is a one-to-one and onto mapping from one space to the other that preserves all the properties defining the space.
Any n-dimensional Euclidean space is isomorphic to $\mathbb{R}^n$.
Note: Although two spaces may be isomorphic as Euclidean spaces, perhaps the same two spaces are not isomorphic when viewed as another space.
Given a basis, any vector can be expressed uniquely as a linear combination of the basis elements. For example, if $\boldsymbol{x}$ = $\sum_ix_i\boldsymbol{x_i}$ for some basis $\boldsymbol{x_i}$, one can refer to $x_i$ as the coordinates of $\boldsymbol{x}$ in terms of this basis.