What aspect does a linear transformation not affect?

A linear transformation, by definition, preserves the zero vector, meaning that if you apply a linear transformation to the zero vector of the vector space, it will still map to the zero vector. This property includes maintaining the origin in the vector space, which is crucial for the transformation's linearity. The transformation alters vectors while keeping the zero vector in place, highlighting that it fundamentally does not affect the origin of the space.

In terms of the other aspects mentioned, a linear transformation does affect the fundamental structure of the vector space because applying it can change how vectors relate to one another within that space, although they will continue to adhere to linearity. Similarly, it affects the relative positions of vectors because a transformation can change the way vectors are aligned or oriented in the new space. Finally, the dimensionality of the vector space can be influenced by the transformation as well; for instance, linear transformations can map a higher-dimensional space into a lower-dimensional one or vice versa, depending on how they are defined.