Transformations

Linearity of Transformations

Linearity of transformation is a different concept compared to linearity of function. For ex,$y=mx+c$ is linear function but not a linear transformation. Since it doesn't hold the laws for linear transformation as the function has a constant term added, in fact, it's an affine transformation.

Mathematically, a transformation $T$ is linear if

$$T(u+v) = T(u)+T(v) \\ T(\alpha u ) = \alpha T(u)$$

Origin has to be part of linear transformation, else the linear properties won't be held.

Eigen Vectors & Values

Eigen vectors and their correspdonging eigen values basically are the vectors in the space, such that when the transformation is applied on them, the vectors just scales instead of actually moving. The scaling constant of that vector is called it's eigen value.

• There are supposed to be as many eigen values as the dimension of the space.