What is a linear transformation?

A linear transformation is fundamentally defined as a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means that if you take two vectors and apply the linear transformation, the resultant output should be the same as if you had added the two vectors together first and then applied the transformation. Additionally, when you scale a vector by a scalar and then apply the transformation, the result should be the same as if you had applied the transformation first and then scaled the output.

For instance, if ( T ) is a linear transformation, then it holds that for any vectors ( u ) and ( v ) in the vector space and any scalar ( c ):

1. ( T(u + v) = T(u) + T(v) )
2. ( T(cu) = cT(u) )

This property is what characterizes a linear transformation. The other choices provided do not correctly encapsulate this defining feature of linear transformations:

• A function that modifies the size of vectors does not necessarily preserve the essential linear properties required for a linear transformation.
• A transformation that only results in zero vectors describes a specific case (the zero transformation) and does not generalize to all linear transformations, which can produce