The linear operator T: R2→R2 defined by the equations
w1 = 4×1 – 6×2
w2 = -2×1 + 3×2
is not one-to-one. Using the methods in class, show why this is true. Once you have done this, provide a simple, specific, numerical example, where the output vector is not the zero vector, that illustrates why the transformation is not one-to-one.
Okay, so I was able to answer the first part (proving the transformation is not one to one).
But for the second part, I'm not exactly sure what to do. Do I just give any input vector that produces a nonzero output?
Best Answer
One-to-one transformations have the property that if, for some vectors $u, v$, $T(u) = T(v)$, $u = v$. Therefore, to show that this transformation is not one-to-one, you should provide an example of two vectors $u, v$ such that $T(u) = T(v)$. The zero vector condition simply means that $T(u) \neq 0$.