Question

Describe all least-squares solutions of the system.

\(\begin{aligned}{}x + y &= 2\\x + y &= 4\end{aligned}\)

Short answer

The solution is the set of all \(\left( {x,y} \right)\) such that \(x + y = 3\).

Step-by-step solution

System of equations in matrix form

The matrix form of system of equations is as follows:

\(\begin{aligned}{}Ax &= b\\\left( {\begin{aligned}{{}}1&1\\1&1\end{aligned}} \right)\left( {\begin{aligned}{{}}x\\y\end{aligned}} \right) &= \left( {\begin{aligned}{{}}2\\4\end{aligned}} \right)\end{aligned}\)

Compute the normal equations

The normal equation has the form of \(\left( {{A^T}A} \right)x = {A^T}b\).

For the given case, the normal equation will be:

\(\begin{aligned}{}\left( {\left( {\begin{aligned}{{}}1&1\\1&1\end{aligned}} \right)\left( {\begin{aligned}{{}}1&1\\1&1\end{aligned}} \right)} \right)\left( {\begin{aligned}{{}}x\\y\end{aligned}} \right) &= \left( {\left( {\begin{aligned}{{}}1&1\\1&1\end{aligned}} \right)\left( {\begin{aligned}{{}}2\\4\end{aligned}} \right)} \right)\\\left( {\begin{aligned}{{}}2&2\\2&2\end{aligned}} \right)\left( {\begin{aligned}{{}}x\\y\end{aligned}} \right) &= \left( {\begin{aligned}{{}}6\\6\end{aligned}} \right)\end{aligned}\)

Above matrix can be converted into system of equation as follows:

\(\begin{aligned}{}2x + 2y = 6\\2x + 2y = 6\end{aligned}\)

Which gives the solution as \(x + y = 3\).

Hence, the solution is the set of all \(\left( {x,y} \right)\) such that \(x + y = 3\).