
Linear Operators
Show that the range of the linear operator defined by the equations
w1 = 4x1  2x2
w2 = 2x1  x2
is not all of R2, and find a vector that is not in the range.
Ok I can easily show the range is not all of R2 by finding the determinant to be zero... but how the vector which is not in the range? I know this is probably really easy.

Hi
$\displaystyle
w_1 =4x_12x_2$ and $\displaystyle
w_2 =2x_1x_2
\implies
w_1=2w_2
\implies\begin{pmatrix}w_1\\w_2 \end{pmatrix}= w_2\begin{pmatrix}2\\1 \end{pmatrix} $
All the vectors of the range can be written as $\displaystyle \lambda \begin{pmatrix}2\\1 \end{pmatrix}$ hence, if you manage to find a vector of $\displaystyle \mathbb{R}^2$ which direction is not that of $\displaystyle \begin{pmatrix}2\\1 \end{pmatrix}$, you are sure it does not belong to the range of the operator.

Okay I get it, thanks so much :)