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Linear Vector Spaces: Change of Basis

In this section, we will introduce the concept of transformation between coordinate systems. The components of vectors in a linear vector spaces depend on the chosen basis set of vectors. If an alternate basis set is chosen, the components of the vector change according to a transformation matrix that is calculated based on the relationship between the vectors in the initial basis set and those in the alternate basis set. Before the coordinate transformation matrices are introduced, we will introduce the Kronecker delta ( $\delta_{ij}$ ), which is a tool that will shorten the derivations below. By $\delta_{ij}$ we mean the following function of $i$ and $j$ :

$\delta_{ij}=\begin{cases} 0& \text{ if }i\neq j\\ 1& \text{ if }i= j \end{cases}$

Let $B=\{e_1,e_2,e_3,\cdots,e_n\}$ be an orthonormal set of basis vectors in $\mathbb{R}^n$ . Then, $\forall u \in \mathbb{R}^n:\exists \{u_1,u_2,u_3,\cdots,u_n\}\subset \mathbb{R}$ such that:

$u=u_1e_1+u_2e_2+u_3e_3+\cdots+u_ne_n$

The numbers $\{u_i\}_{i=1}^n$ are called the components of $u$ in the orthonormal basis set $B$ and each component can be obtained using the dot product operation:

$u_i = u\cdot e_i$

Then, the vector $u$ can also be written as:

$u=(u\cdot e_1)e_1+(u\cdot e_2)e_2+(u\cdot e_3)e_3+\cdots+(u\cdot e_n)e_n=\sum_{j=1}^n(u\cdot e_j)e_j$

Let $B'=\{e'_1,e'_2,e'_3,\cdots,e'_n\}$ be a different orthonormal set of basis vectors in $\mathbb{R}^n$ , and let $u'$ be the representation of the vector $u$ in the new coordinate system. In this case, we have a new set of components $\{u'_1,u'_2,u'_3,\cdots,u'_n\}\subset \mathbb{R}$ such that:

$u'=u'_1e'_1+u'_2e'_2+u'_3e'_3+\cdots+u'_ne'_n$

In order to find the relationship between the components $\{u_i\}_{i=1}^n$ and $\{u'_i\}_{i=1}^n$ we first find the relationship between the basis vectors. As a first step, each basis vector $e'_i$ will be expressed in terms of the original basis vectors $\{e_1,e_2,e_3,\cdots,e_n\}$ as was done above for $u$ :

$e'_i=(e'_i\cdot e_1)e_1+(e'_i\cdot e_2)e_2+(e'_i\cdot e_3)e_3+\cdots+(e'_i\cdot e_n)e_n=\sum_{j=1}^n(e'_i\cdot e_j)e_j$

Let $Q$ denote the matrix of numbers $Q_{ij}=e'_i\cdot e_j$ . This matrix of numbers is the matrix of transformation between the components in the basis sets $B$ and $B'$ . Let $Q^T$ denote the transpose of $Q$ , i.e.:

$(Q^T)_{ij}=Q_{ji}$

We will now show that the rows of $Q$ and $Q^T$ are orthonormal.
Indeed, since we know that the set $B'=\{e'_1,e'_2,e'_3,\cdots,e'_n\}$ is orthonormal, we have:

$\begin{split} \delta_{ij}&=e'_i\cdot e'_j=\left(\sum_{k=1}^n(e'_i\cdot e_k)e_k\right) \cdot \left(\sum_{l=1}^n(e'_j\cdot e_l)e_l\right)\\ &=\sum_{k,l=1}^n(Q_{ik}Q_{jl})\delta_{kl}\\ &=\sum_{k=1}^n(Q_{ik}Q_{jk}) \end{split}$

i.e., if we multiply the components of the $i^{th}$ row of $Q$ by the components of the $j^{th}$ row of $Q$ , we get $0$ when $i\neq j$ and we get $1$ when $i=j$ .

If we use $I$ to denote the identity matrix (i.e., with 1 in the diagonal components and 0 in the off diagonal components $I_{ij}=\delta_{ij}$ ) then we have:

$I=QQ^T$

By expressing the basis vectors $\{e_i\}_{i=1}^n$ in terms of the orthonormal basis set of vectors $B'=\{e'_1,e'_2,e'_3,\cdots,e'_n\}$ and using the same argument above we have:

$I=Q^TQ$

Thus, the matrix $Q$ is an example of a special type of operators (orthogonal matrices) which will be studied later. This matrix can be used to relate the components of $u$ with the components of $u'$ as follows:

$\begin{split} u'_i & =u\cdot e'_i=\sum_{j=1}^n (u_je_j)\cdot e'_i=\sum_{j=1}^n (e'_i\cdot e_j)u_j\\ & =\sum_{j=1}^nQ_{ij}u_j \end{split}$

This last relationship can be written in matrix form as follows. If $u$ represents the components of a vector in the orthonormal basis set $B$ and $u'$ represents the components of that vector in the orthonormal basis set $B'$ , then, these components are related by the matrix $Q$ :

$u'=Qu$

The webMathematica tools in the Matrix Representation and Change of Basis is a useful for the illustration of the concept and calculations of coordinate transformations in 2D and 3D.

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Linear Vector Spaces: Change of Basis

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