Which operation is preserved in a linear transformation?

In a linear transformation, the operation that is preserved is vector addition. This means that if you apply a linear transformation to the sum of two vectors, the result is the same as if you applied the transformation to each vector individually and then added the transformed vectors together.

Mathematically, if ( T ) is a linear transformation and ( \mathbf{u} ) and ( \mathbf{v} ) are vectors in the vector space, the property of linearity can be stated as follows:

[ T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) ]

This property is fundamental to linear transformations and ensures that they maintain the structure of vector addition. In the context of linear algebra and finite mathematics, understanding this preservation of operations is crucial, as it underpins much of the analysis involving vector spaces.

The other operations such as vector division, vector subtraction, and scalar averaging do not hold the same linearity property in the context of linear transformations. For example, while vector subtraction can be related to vector addition (as it is simply adding the negative of a vector), it is not an independent operation preserved under linear transformations the same way vector addition is.