Math 511: Linear Algebra

QR Factorization

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1. Find the QR Factorization of each matrix.¶

(a) $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \\ 1 & 0 \end{bmatrix}$

(b) $A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \\ 1 & 1 \\ 1 & 2 \end{bmatrix}$

2. Let $A=QR$ be the QR-Factorization of the $m\times n$ matrix $A$ of rank $n$. Show how the least squares problem can be solved using the QR-Factorization.¶

3. Use the result of part 2 to solve the least squares problem¶

$A\mathbf{x} = \mathbf{b}$ when $A$ is the matrix from 1.(a) and $\mathbf{b} = \begin{bmatrix}-1\ \\ \ \ 1\ \\ -1 \end{bmatrix}$.

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