### Linear Transformation

##### 4. Linearity in picture

We can often visualize transformations between euclidean spaces. For example, the name of transformation Reflection in x-axis: R2R2, (x, y) → (x, -y) is inspired by the picture.

Example The transformation Reflection in y-axis: R2R2, (x, y) → (-x, y)

is given by the matrix

 [ -1 0 ] 0 1

The transformation Rotation by &\theta;: R2R2, (x, y) → (xcosθ + ysinθ, - xsinθ + ycosθ)

is given by the matrix (remember writing vectors vertically)

 [ cosθ - sinθ ] sinθ cosθ

The transformation Projection to (x1, x3)-plane: R3R2, (x1, x2, x3) → (x1, x3)

is given by the matrix

 [ 1 0 0 ] 0 0 1

Linear transformations are characterized by the property that addition and scalar multiplication are preserved. Geometrically, this means that parallelograms and stretching/shrinking are preserved.

Example The transformation Reflection in x-axis clearly preserves parallelograms and stretching/shrinking.

The rotation transformation also preserves the two operations.

The projection transformation is also a linear transformation.