Transformation of vector components

X.B.2.1 Transformation of vector components

Consider a vector $v$ with components expressed in two Cartesian coordinate systems with base vectors $e_{i}$ and $e_{i}^{'}$ respectively. Thus:

v = v_{i} e_{i} = v_{i}^{'} e_{i}^{'} .

In the last expression we introduce the notations $v_{i}$ and $v_{i}^{'}$ for the components of vector $v$ in $e_{i}$ and $e_{i}^{'}$ respectively.

It is possible to decompose the vectors $e_{i}^{'}$ as a linear combination of vectors $e_{j}$ :

e_{i}^{'} = A_{i j} e_{j} .

The coefficients $A_{i j}$ are easily calculated. Indeed, the scalar multiplication of the previous equality by $e_{k}$ gives successively:

\begin{align} e_{i}^{'} \cdot e_{k} & = (A_{i j} e_{j} \cdot e_{k}), \\ e_{i}^{'} \cdot e_{k} & = (A_{i j} δ_{j k}), \\ e_{i}^{'} \cdot e_{k} & = A_{i k}, \end{align}

or finally:

A_{i j} = e_{i}^{'} \cdot e_{j} .

By a similar calculation, it is possible to identify the relations between the components of vector $v$ expressed in the two coordinate systems:

\begin{align} e_{k}^{'} \cdot v = e_{k}^{'} \cdot (v_{i} e_{i}) & = e_{k}^{'} \cdot (v_{i}^{'} e_{i}^{'}), \\ v_{i} (e_{k}^{'} \cdot e_{i}) & = v_{i}^{'} (e_{k}^{'} \cdot e_{i}^{'}), \\ v_{i} A_{k i} & = v_{i}^{'} δ_{k i}, \\ A_{k i} v_{i} & = v_{k}^{'}, \end{align}

Finally, it can be seen that the relation between base vectors and vector components are the same:

e_{i}^{'} = A_{i j} e_{j},

v_{i}^{'} = A_{i j} v_{j} .

This is why the transformation is called a covariant transformation. The quantities $v_{i}$ and $v_{i}^{'}$ are called covariant components of vector $v$ in coordinate systems $e_{i}$ and $e_{i}^{'}$ respectively. In the last vector component transformation one recognizes a classical algebraic result:

V^{'} = A \cdot V .

The matrix $A_{i j}$ is orthogonal: ${(A_{i j})}^{- 1} = A_{j i}$ and the reverse relation for vector components is:

v_{j} = A_{j i} v_{i}^{'} .

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