Linear Algebra with Applications, 10th Edition

3.5 Change of Basis

Many applied problems can be simplified by changing from one coordinate system to another. Changing coordinate systems in a vector space is essentially the same as changing from one basis to another. For example, in describing the motion of a particle in the plane at a particular time, it is often convenient to use a basis for ${ℝ}^2$ consisting of a unit tangent vector t and a unit normal vector n instead of the standard basis $\{{\mathbf{e}}_1,{\mathbf{e}}_2\}$.

In this section, we discuss the problem of switching from one coordinate system to another. We will show that this can be accomplished by multiplying a given coordinate vector x by a nonsingular matrix S. The product $\mathbf{y}=S\mathbf{x}$ will be the coordinate vector for the new coordinate system.

Changing Coordinates ...