Which of the following describes a linear transformation?

A linear transformation is defined as a mapping between two vector spaces that preserves the operations of addition and scalar multiplication. This means that if you take two vectors and add them together or multiply a vector by a scalar, the transformation will produce the same results as if you had applied the transformation to each vector first and then performed the addition or scalar multiplication. This property is essential for maintaining the structure of the vector spaces involved.

The other choices do not adequately describe a linear transformation. For instance, claiming that a linear transformation can only transform vectors in three-dimensional space restricts its applicability since linear transformations can operate in any finite-dimensional vector space. Additionally, stating that it is non-continuous contradicts the definition of linear transformations, which are inherently continuous due to their linear nature. Lastly, the assertion that linear transformations can only operate on square matrices is incorrect, as linear transformations can apply to vectors and matrices of varying dimensions, not just square ones. Thus, the first choice accurately encapsulates the fundamental characteristics of linear transformations.