1.2 Matrices as linear maps

Work through these

• Matrix–vector product as a change of coordinates

  Multiplying a matrix by a vector is best read as a transformation of that vector rather than as a rule for arranging numbers. Once you see the picture, matrix notation becomes descriptive instead of arbitrary.

• Composition of maps = matrix multiplication

  Applying one transformation after another is exactly what multiplying two matrices does, which is why the order matters and why the shapes have to line up. This single fact explains most matrix algebra rules.

• Special matrices: diagonal, orthogonal, symmetric, sparse

  Diagonal, orthogonal, symmetric and sparse matrices each have a geometric meaning worth attaching to the name. Recognising them tells you what a computation will do before you run it.

• Broadcasting rules you will meet again in NumPy

  The rules for combining arrays of different shapes appear first in the mathematics and again in the numerical libraries. Learning them here saves a great deal of confusion later.