# Thread: True or false linear transformation

1. ## True or false linear transformation

Are the following true or false? If it is true, prove it, if it is false give a counter example (and proof that your counter example is true).

1) If T is a linear transformation such that T: $R^2$ -> $R^3$, T can't be one to one.

2) If a matrix has characteristic polynomial $f(x) = (x-2)^2(x+4)(x-4)$, then the matrx is not diagonalizable.

2. Originally Posted by cindy
Are the following true or false? If it is true, prove it, if it is false give a counter example (and proof that your counter example is true).

1) If T is a linear transformation such that T: $R^2$ -> $R^3$, T can't be one to one.
$T(x,y)=(x,y,0)$ is one to one.
May be you should ask $T:\mathbb{R}^{3}\to\mathbb{R}^{2}$?

3. Originally Posted by cindy
1) If T is a linear transformation such that T: $R^2$ -> $R^3$, T can't be one to one.
Like bkarpuz said it should be the other way around if you want to have an impossibility.
In general $T: \mathbb{R}^n \to \mathbb{R}^m$ is not one-to-one if $n>m$ by rank-nullity theorem.

2) If a matrix has characteristic polynomial $f(x) = (x-2)^2(x+4)(x-4)$, then the matrx is not diagonalizable.
$\begin{bmatrix} 2&0&0&0\\0&2&0&0\\0&0&-4&0\\0&0&0&4 \end{bmatrix}$