A linear map is a mathematical function or transformation that preserves the operations of addition and scalar multiplication. In other words, the output of a linear map when applied to a linear combination of inputs will be the same as the linear combination of the outputs of the map applied to each individual input.
The dimension of the image (Im) of a linear map T refers to the number of linearly independent vectors in the range of the map. In this case, dim(Im(T))=k means that the image has a basis of k linearly independent vectors. This is important because it tells us the size and structure of the image of the linear map.
The rank of a linear map T is defined as the dimension of the image of T. Therefore, if dim(Im(T))=k, then the rank of T is also k.
Yes, a linear map can have a dimension of 0. This means that the image of the map is the zero vector (or the trivial vector space). In other words, the map is not able to map any vectors to a nonzero output.
The nullity of a linear map T is defined as the dimension of the kernel (null space) of T. The null space is the set of all vectors that are mapped to the zero vector by T. Therefore, if dim(Im(T))=k, then the nullity of T is n-k, where n is the dimension of the domain of T.