关于线性变换教学的注记

Notes on the Teaching of Linear Transformation

线性变换与它在一组基下的矩阵是一一对应的。通过具体的实例,我们强调线性变换作用于向量,通常不等于对应的矩阵和这个向量的乘积;矩阵乘以此向量在基下的坐标作成的向量,等于变换后的向量在同一组基下的坐标作成的向量;在特殊情形下,线性变换作用于向量,等于对应的矩阵与此向量的乘积,这是个有实用性的结果。

The linear transformation has a one-to-one correspondence with its matrix under a set of bases.In this note,through specific examples,we emphasize that the linear transformation applying to a vector is usually not equal to the product of the corresponding matrix and the vector;The matrix multiplied by the vector made by the coordinates of this vector under the chosen base is equal to the vector made by the coordinates of the transformed vector under the same base;Under special circumstances,the linear transformation acting on the vector equals to the product of the corresponding matrix and this vector,which is a practical result.