Linear transformations and matrices

What makes a transformation "linear"?

• Transformation of summed vectors, is equal to the sum of their individual transformation.

• And scaling preserved

Matrics

• We use matrics to describe linear transformations numerically.

• The effect of the transformation on all vectors can be summarized by what the transformation does the unit vectors of $\hat{i},\hat{j}$

• By knowing what the transformation does to unit vectors, we can extend it to all other vectors.

Example

Suppose you get the data descibing what happens to $\hat{i},\hat{j}$

Extend this transformation to some vector, $\vec{v} = \begin{pmatrix} -1 \\ 2 \end{pmatrix}$

How can you then define linear transformation generically:

Matrix Multiplication

• https://www.3blue1brown.com/lessons/matrix-multiplication

• Non-commutative -> AB != BA

• Associative -> A(BC) = AB(C)

Order of transformation matters!

1. Applying transformation B then A, would be different from applying transformation A then B. Hence non-commutative.

2. Applying transformation C, then B, finally A; brackets do not alter the order of transformation. Therefore, associative.