The span of a set of linear transformations is the set of all possible linear combinations of those transformations. In other words, it is the space that can be reached by transforming any vector in the original set.
To calculate the span, you can use the row reduction method or the column space method. The row reduction method involves putting the transformations into a matrix and reducing it to its row echelon form. The nonzero rows in the resulting matrix will form a basis for the span. The column space method involves taking the columns of the matrix and finding the linear combinations that result in the zero vector. The resulting vectors will form a basis for the span.
Understanding the span of a set of linear transformations is important because it allows us to determine the range and dimension of the transformations. It also helps us to understand the transformation better and make predictions about how it will affect vectors in the original set.
Yes, the span of a set of linear transformations can be greater than the original set. This can happen when the transformations are linearly dependent, meaning that one or more of the transformations can be expressed as a linear combination of the others. In this case, the span will contain redundant vectors.
The span of a set of linear transformations will be equal to the original set if and only if the transformations are linearly independent. If the transformations are linearly dependent, the span will be larger than the original set. Therefore, understanding the span can also help us determine the linear independence of a set of transformations.