Matrix on Vector Math

A matrix can be used to calculate a new vector from an old vector. In our notation the prime symbol, (‘), will indicate new.

More specifically, each component in the new vector \overrightarrow{v'} is a sum of contributions from all components of the old vector \overrightarrow{v}.

New Vector Components

v' = \begin{bmatrix} v'_1 \\ v'_2 \\ v'_3 \end{bmatrix}

v'_1 = a_{11} v_1 + a_{12} v_2 + a_{13} v_3

v'_2 = a_{21} v_1 + a_{22} v_2 + a_{23} v_3

v'_3 = a_{31} v_1 + a_{32} v_2 + a_{33} v_3

If you have worked dozens of matrix multiplication problems, you might notice that the three equations above show what we would get from the following matrix multiplication:

\begin{bmatrix} a_{11} && a_{12} && a_{13} \\ a_{21} && a_{22} && a_{23} \\ a_{31} && a_{32} && a_{33} \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} = \begin{bmatrix} v'_1 \\ v'_2 \\ v'_3 \end{bmatrix}

The components of both vectors (old and new) must be the same type of mathematical thing. The components of the a matrix must be Scalars.