Linear Transformation and Its Stand Matrix

This video states what is a linear transformation and how to find its standard matrix. The linear transformation is defined as a transformation which is closed under addition and scalar multiplication. That is, \(T(\vec{u}+\vec{v})=T(\vec{u})+T(\vec{v})\) and \(T(\lambda \vec{u})=\lambda T(\vec{u})\) for all \(\lambda\in\mathbb{R}\).

To find the standard matrix of a linear transformation, we only need to find the images of all vectors in the standard basis. Arranging the image vectors as the columns of a matrix,this matrix is the standard matrix of the linear transformation.