Determine when vector is not in range of Matrix Transformation

• Feb 9th 2009, 05:51 AM
jennifer1004
Determine when vector is not in range of Matrix Transformation
Let $R^2->R^2$be the matrix transformation defined by f(x)=Ax, where $A=\begin{pmatrix}2 & 1 & 2\\1 & 0 & -1\\3 & 1 & k\end{pmatrix}$

Determine k so that $w=\begin{pmatrix}1\\1\\1\end{pmatrix}$ is not in the range of f.
• Feb 9th 2009, 07:06 AM
abender
Quote:

Originally Posted by jennifer1004
Let $R^2->R^2$be the matrix transformation defined by f(x)=Ax, where $A=\begin{pmatrix}2 & 1 & 2\\1 & 0 & -1\\3 & 1 & k\end{pmatrix}$

Determine k so that $w=\begin{pmatrix}1\\1\\1\end{pmatrix}$ is not in the range of f.

We want to determine k so that the column vector $w=\begin{pmatrix}1\\1\\1\end{pmatrix}$ is NOT in the range of f.

So, we want k such that for any vector x, $Ax = \begin{pmatrix}1\\1\\1\end{pmatrix}$ NEVER HOLDS.

That is, $\begin{pmatrix}2 & 1 & 2\\1 & 0 & -1\\3 & 1 & k\end{pmatrix} x = \begin{pmatrix}1\\1\\1\end{pmatrix}$ NEVER HOLDS.

How can we assure this? What must k be/not be?