Linear Algebra.Linear Transformation.Help

**ypatia** · March 5th 2009, 09:50 AM

Suppose T is the linear transformation on R^3 that takes each point (x,y,z) to (x+y+z, x+y,z). Describe what T^-1 does to the point (x,y,z). Thanks in advance.

**Grandad** · March 5th 2009, 11:25 AM

Hello ypatia

Originally Posted by ypatia

Suppose T is the linear transformation on R^3 that takes each point (x,y,z) to (x+y+z, x+y,z). Describe what T^-1 does to the point (x,y,z). Thanks in advance.

Do you know how to find the inverse of a 3x3 matrix? There are quite a few web-sites that will explain the method, if you don't. Just Google 'Inverse of a 3x3 matrix'.

In the question you have here, the transformation can be written in matrix form as

$\begin{pmatrix}1&1&1\\1&1&0\\1&0&0\end{pmatrix}\be gin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}x+y+z\\x+y\\z\end{pmatrix}$

So, to find the inverse transformation, you'll need the inverse of the matrix $\begin{pmatrix}1&1&1\\1&1&0\\1&0&0\end{pmatrix}$ , which is $\begin{pmatrix}0&0&1\\0&1&-1\\1&-1&0\end{pmatrix}$

To find its effect on $(x, y, z)$ , just do the matrix multiplication:

$\begin{pmatrix}0&0&1\\0&1&-1\\1&-1&0\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix } = \begin{pmatrix}z\\y-z\\x-y\end{pmatrix}$

So there's your answer: $T^{-1}<img src=$ x,y,z) \rightarrow (z,y-z,x-y)" alt="T^{-1}

x,y,z) \rightarrow (z,y-z,x-y)" />

Grandad

**ypatia** · March 5th 2009, 11:43 AM

Originally Posted by Grandad

Hello ypatiaDo you know how to find the inverse of a 3x3 matrix? There are quite a few web-sites that will explain the method, if you don't. Just Google 'Inverse of a 3x3 matrix'.

In the question you have here, the transformation can be written in matrix form as

$\begin{pmatrix}1&1&1\\1&1&0\\1&0&0\end{pmatrix}\be gin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}x+y+z\\x+y\\z\end{pmatrix}$

So, to find the inverse transformation, you'll need the inverse of the matrix $\begin{pmatrix}1&1&1\\1&1&0\\1&0&0\end{pmatrix}$ , which is $\begin{pmatrix}0&0&1\\0&1&-1\\1&-1&0\end{pmatrix}$

To find its effect on $(x, y, z)$ , just do the matrix multiplication:

$\begin{pmatrix}0&0&1\\0&1&-1\\1&-1&0\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix } = \begin{pmatrix}z\\y-z\\x-y\end{pmatrix}$

So there's your answer: $T^{-1}<img src=$ x,y,z) \rightarrow (z,y-z,x-y)" alt="T^{-1}

x,y,z) \rightarrow (z,y-z,x-y)" />

Grandad

Is this right?? "T is not injective and, as such, it does not have an inverse. For an example of the lack of injectivity, T(1,0,1) -->(2,1,1) and T(2,-1,1) -->(2,1,1)." Because if this is right contradicts to the above solution of T^-1. Thanks in advance