A linear transformation is a function that maps one vector space to another in a way that preserves its structure. In simpler terms, it is a mathematical operation that takes in a vector and outputs another vector, while following certain rules.
"im(S+T)" refers to the image or range of the linear transformation that results from adding two linear transformations, S and T, together. It represents all possible outputs that can be obtained by applying the combined transformation to the input vectors.
It is possible for im(S+T) to be a subset of im(S) + im(T), meaning that all outputs of the combined transformation can be obtained by separately applying the individual transformations of S and T and then adding their respective outputs together. However, in some cases, im(S+T) may be equal to im(S) + im(T).
No, it is not possible for im(S+T) to be larger than im(S) + im(T). This is because im(S+T) must be a subset of im(S) + im(T), meaning that it can only contain outputs that are possible from applying both S and T separately.
Linear transformations have many practical applications, such as in computer graphics, signal processing, and data analysis. For example, they can be used to rotate and scale images, filter out noise from signals, and reduce the dimensionality of data sets for easier analysis.