Prove linear transformation is one to one

Question

The linear operator T: R2→R2 defined by the equations

w1 = 4x1 - 6x2

w2 = -2x1 + 3x2

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?

Answer 1

You are suppose to find $x_1, x_2, y_1, y_2$ such that $(x_1, x_2) \neq (y_1, y_2), (w_1, w_2) \ne (0,0),$ and

$$4x_1-6x_2=w_1 = 4y_1-6y_2$$

$$-2x_1+3x_2=w_2 = -2y_1+3y_2$$

Answer 2

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 $T(u) \neq 0$.