Define the linear least squares problem for the equation

$$A \mathbf{x}=\mathbf{b}$$

where $A$ is a given $m \times n$ matrix with $m>n, \mathbf{b} \in \mathbb{R}^{m}$ is a given vector and $\mathbf{x} \in \mathbb{R}^{n}$ is an unknown vector.

Explain how the linear least squares problem can be solved by obtaining a $Q R$ factorization of the matrix $A$, where $Q$ is an orthogonal $m \times m$ matrix and $R$ is an uppertriangular $m \times n$ matrix in standard form.

Use the Gram-Schmidt method to obtain a $Q R$ factorization of the matrix

$$A=\left(\begin{array}{lll} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right)$$

and use it to solve the linear least squares problem in the case

$$\mathbf{b}=\left(\begin{array}{l} 1 \\ 2 \\ 3 \\ 6 \end{array}\right)$$