In a three-dimensional space the coordinate system defined by the base vectors a', b', c' may be defined in terms of the base vectors a, b, c by three equations (supposing the transformation leaves the origin invariant):

where m_ij are any real numbers. In matrix notation it is written as:

The reverse transformation will be A = M ^-1 A'. The vector:

may be written in the new coordinate system A' as:

Introducing the above expressions of  a', b', and c' into the r' expression leads to:

The reverse transformation will be:

It is important to note that the above expression provides the transformation rule for the components of the vector r while the vector itself is unaffected (r' = r) by the change of axes.