Linear Operators

• May 18th 2008, 11:33 PM
steph_b23
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.
• May 19th 2008, 08:24 AM
flyingsquirrel
Hi

$\displaystyle w_1 =4x_1-2x_2$ and $\displaystyle w_2 =2x_1-x_2 \implies w_1=2w_2 \implies\begin{pmatrix}w_1\\w_2 \end{pmatrix}= w_2\begin{pmatrix}2\\1 \end{pmatrix}$

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.
• May 19th 2008, 09:21 AM
steph_b23
Okay I get it, thanks so much :)