The rank of a linear transformation is the dimension of the vector space spanned by the range of the transformation. In other words, it is the number of linearly independent columns or rows in the matrix representation of the transformation.
The rank of a linear transformation plus its nullity (the dimension of the null space) always equals the dimension of the domain of the transformation. This is known as the rank-nullity theorem and is a fundamental property of linear transformations.
Yes, the rank of a linear transformation can change depending on the choice of basis for the vector space. However, the maximum possible rank of a transformation is always fixed and is equal to the smaller of the dimensions of the domain and range of the transformation.
The rank of a linear transformation can be computed by performing row reduction on the matrix representation of the transformation and counting the number of non-zero rows in the reduced matrix. Alternatively, the rank can also be found by counting the number of pivot positions in the matrix.
A higher rank of a linear transformation indicates that the transformation maps a larger portion of the domain to the range, and thus has a more diverse set of outputs. This can be useful in applications such as data compression and image processing, where a higher rank transformation can preserve more information and result in better quality results.