# 6.2: Linear Functions on Hyperplanes

- Page ID
- 1899

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It is not always so easy to write a linear operator as a matrix. Generally, this will amount to solving a linear systems problem. Examining a linear function whose domain is a hyperplane is instructive.

Example 63

Let $$V=\left\{ c_{1}\begin{pmatrix}1\\1\\0\end{pmatrix} +c_{2}\begin{pmatrix}0\\1\\1\end{pmatrix} \middle| c_{1},c_{2}\in \Re \right\} $$ and consider \(L:V\to \Re^{3}\) defined by

$$

L\begin{pmatrix}1\\1\\0\end{pmatrix} = \begin{pmatrix}0\\1\\0\end{pmatrix},\qquad

L\begin{pmatrix}0\\1\\1\end{pmatrix} = \begin{pmatrix}0\\1\\0\end{pmatrix}.

$$

By linearity this specifies the action of \(L\) on any vector from \(V\) as

$$

L\left[ c_{1}\begin{pmatrix}1\\1\\0\end{pmatrix} + c_{2} \begin{pmatrix}0\\1\\1\end{pmatrix} \right]= (c_{1}+c_{2})\begin{pmatrix}0\\1\\0\end{pmatrix}.

$$

The domain of \(L\) is a plane and its range is the line through the origin in the \(x_{2}\) direction. It is clear how to check that \(L\) is linear.

It is not clear how to formulate \(L\) as a matrix;

since

\begin{eqnarray*}

L\begin{pmatrix}c_{1}\\\!\!c_{1}+c_{2}\!\!\\c_{2}\end{pmatrix} =

\begin{pmatrix}

0&0&0\\

1&0&1\\

0&0&0

\end{pmatrix}

\begin{pmatrix}c_{1}\\\!\!c_{1}+c_{2}\!\!\\c_{2}\end{pmatrix} =(c_{1}+c_{2})\begin{pmatrix}0\\1\\0\end{pmatrix}\, ,

\end{eqnarray*}

or since

\begin{eqnarray*}

L\begin{pmatrix}c_{1}\\\!\!c_{1}+c_{2}\!\!\\c_{2}\end{pmatrix} =

\begin{pmatrix}

0&0&0\\

0&1&0\\

0&0&0

\end{pmatrix}

\begin{pmatrix}c_{1}\\\!\!c_{1}+c_{2}\!\!\\c_{2}\end{pmatrix} =(c_{1}+c_{2})\begin{pmatrix}0\\1\\0\end{pmatrix}

\end{eqnarray*}

you might suspect that \(L\) is equivalent to one of these \(3\times3\) matrices. It is not. All \(3\times3\) matrices have \(\Re^{3}\) as their domain, and the domain of \(L\) is smaller than that. When we do realize this \(L\) as a matrix it will be as a \(3\times2\) matrix. We can tell because the domain of \(L\) is 2 dimensional and the codomain is \(3\) dimensional

## Contributor

David Cherney, Tom Denton, and Andrew Waldron (UC Davis)