What kind of operations does a transformation matrix typically allow?

A transformation matrix typically allows both addition and scalar multiplication of vectors in the context of vector spaces. When applying a transformation matrix to a vector, the operation corresponds to a linear transformation, which adheres to the principles of linearity. This means that for a transformation matrix ( T ), if ( v ) and ( w ) are vectors, and ( c ) is a scalar, the following two properties hold:

1. Additivity: The transformation of the sum of vectors is equal to the sum of the transformations of each vector. That is, ( T(v + w) = T(v) + T(w) ).

2. Scalar Multiplication: The transformation of a scalar multiple of a vector is equal to the scalar multiple of the transformation of that vector. That is, ( T(cv) = cT(v) ).

These properties imply that the transformation matrix operates in a way that preserves the essential structure of vectors within a vector space, allowing for operations such as addition and scalar multiplication to be meaningful and compatible under the transformation.

The other options do not accurately represent the capabilities of a transformation matrix. For instance, stating that transformation matrices only allow for the addition of vectors excludes the fundamental aspect of scalar