math111_logo 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.

reflection in yaxis

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

reflection in x-axis

is given by the matrix

[ -1 0 ]
0 1

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

rotation

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)

projection

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.

reflection is linear

The rotation transformation also preserves the two operations.

rotation is linear

The projection transformation is also a linear transformation.

projection is linear


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