Find the least-squares solutions x⃗^* of the system A x⃗=b⃗, where A=[ 1 2 3 4

step 1 For this exercise, we have a transpose times matrix A equal to 66, 78, 90, 78, 93, 108, 90, 108,

Find the least-squares solutions x⃗^* of the system A x⃗=b⃗,...

Find the least-squares solutions $\vec{x}^{*}$ of the system $A \vec{x}=\vec{b},$ where
$$A=\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right] \quad \text { and } \quad \vec{b}=\left[\begin{array}{l}
1 \\
0 \\
0
\end{array}\right]$$

Key Concepts

Null Space and Minimum Norm Solutions

In cases where there are infinitely many least squares solutions due to the nontrivial null space of the matrix, the concept of the null space becomes important. The minimum norm solution is the unique solution among all least squares solutions that has the smallest Euclidean norm, and it is often preferred for its optimality and simplicity.

Moore-Penrose Pseudoinverse

The Moore-Penrose pseudoinverse generalizes the concept of an inverse to non-square or singular matrices. It provides a method to compute a unique least-squares solution, especially when the matrix A does not have full rank, ensuring that the computed solution minimizes the norm among all possible solutions.

Least Squares Problem

The least squares problem involves finding an approximate solution to an overdetermined or inconsistent system of equations by minimizing the sum of the squares of the residuals (the differences between the observed and computed values). This concept is fundamental in numerical linear algebra and statistics, as it provides a way to obtain a best-fit solution when an exact solution does not exist.

Normal Equations

Normal equations are derived by setting the gradient of the least squares objective function to zero, leading to the equation A?A x = A?b. Solving these equations gives the least squares solution to the problem. They are central to understanding the computational method for obtaining solutions in the least squares sense.

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