# A FEW BITS OF MATRIX ALGEBRA

• A vector has both direction and magnitude. It is often denoted by boldface letters u, v, T, F... or by an arrow above the letter.
• A unit vector is a vector of magnitude zero and is denoted by the symbols i, j, k, or by "hat" over letter (which can also denote principal vectors).

### Summation conventions:

A repeated (dummy) index i denotes a summation with respect to that index.

### Vector Transformations:

A vector u can be defined in any coordinate system. The vector remains the same regardless of the coordinate system. Matrices can be used to transform a vector from one coordinate system to another coordinate system. A coordinate system in three dimensions with axes x,y,z can be rotated to a new coordinate system with axes x',y',z'. A vector can be expressed in terms of its' components in the two orthogonal coordinate systems with unit base vectors ê:

The direction cosines form a transformation matrix A. For a counter clockwise rotation around the Z axis the transformation matrix A is:

The process can also be reversed. The transformation matrix below is just the transpose of the one above.