# True or false linear transformation

• Oct 4th 2008, 06:34 AM
cindy
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: $\displaystyle R^2$ -> $\displaystyle R^3$, T can't be one to one.

2) If a matrix has characteristic polynomial $\displaystyle f(x) = (x-2)^2(x+4)(x-4)$, then the matrx is not diagonalizable.
• Oct 4th 2008, 07:40 AM
bkarpuz
Quote:

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: $\displaystyle R^2$ -> $\displaystyle R^3$, T can't be one to one.

$\displaystyle T(x,y)=(x,y,0)$ is one to one.
May be you should ask $\displaystyle T:\mathbb{R}^{3}\to\mathbb{R}^{2}$?
• Oct 5th 2008, 08:37 PM
ThePerfectHacker
Quote:

Originally Posted by cindy
1) If T is a linear transformation such that T: $\displaystyle R^2$ -> $\displaystyle R^3$, 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 $\displaystyle T: \mathbb{R}^n \to \mathbb{R}^m$ is not one-to-one if $\displaystyle n>m$ by rank-nullity theorem.

Quote:

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