Lesson 33:The Kernel of a Linear Transformation (4.2, 4.3)Given a linear transformation between two vector spaces, one can ask whetheror not it is one-to-one (defined below). Answering this question can require alot of work. In this section, you will learn about the kernel of a linear trans-formation. It turns out that knowledge about the kernel helps determine if thetransformation is one-to-one.Definition 1:A linear transformationT:V→Wis one-to-oneif for all vectorsv1,v2∈V,T(v1) =T(v2)impliesv1=v2.Equivalently,v1̸=v2impliesT(v1)̸=T(v2).Thus,Tis one-to-one if two distinct vectors inVhave two distinct imagesinWunderT.Example 1:LetT:R2→R2be a linear operator defined byT(x1, x2) =(2x1, x1+x2). Show thatTis one-to-one.
Get answer to your question and much more
Example 2: LetT:R3→Rbe a linear transformation defined byT(x1, x2, x3) =x2.