A linear transformation is a mathematical function that maps vectors from one vector space to another in a way that preserves their linear relationships. In simpler terms, it is a transformation that follows the rules of linear algebra.
Linear transformations are important in many areas of mathematics and science, including geometry, physics, and computer graphics. They allow us to understand and manipulate complex systems by breaking them down into simpler linear relationships.
The defect of a linear transformation refers to the difference in dimension between the kernel (null space) and the image (range) of the transformation. It is essentially the number of dimensions that the transformation fails to map onto the output space.
The defect of a linear transformation can be calculated by finding the dimension of its kernel and subtracting it from the dimension of its input vector space. Alternatively, it can also be calculated by finding the dimension of its image and subtracting it from the dimension of its output vector space.
A high defect in a linear transformation indicates that the transformation is not able to map all vectors from the input space to the output space. This means that there are certain vectors that do not have a corresponding output, leading to a loss of information in the transformation process.