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, $T(\overrightarrow{u}+\overrightarrow{v})=T(\overrightarrow{u})+T(\overrightarrow{v})$ and $T(\lambda \overrightarrow{u})=\lambda T(\overrightarrow{u})$ 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.