ALAFF Additional homework

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Subsection 4.6.1 Additional homework

We start with some concrete problems from our undergraduate course titled "Linear Algebra: Foundations to Frontiers" [27]. If you have trouble with these, we suggest you look at Chapter 11 of that course.

Homework 4.6.1.1.

Consider \(A = \left( \begin{array}{r r} 1 \amp 0 \\ 0 \amp 1 \\ 1 \amp 1 \end{array} \right)\) and \(b = \left( \begin{array}{r r} 1 \\ 1 \\ 0 \end{array} \right) \text{.}\)

Compute an orthonormal basis for \(\Col( A ) \text{.}\)
Use the method of normal equations to compute the vector \(\widehat x \) that minimizes \(\min_x \| b - A x \|_2 \)
Compute the orthogonal projection of \(b \) onto \(\Col( A ) \text{.}\)
Compute the QR factorization of matrix \(A \text{.}\)
Use the QR factorization of matrix \(A \) to compute the vector \(\widehat x \) that minimizes \(\min_x \| b - A x \|_2 \)

Homework 4.6.1.2.

The vectors

\begin{equation*} q_0 = \frac{\sqrt{2}} {2} \left( \begin{array}{r} 1 \\ 1 \\ \end{array} \right) = \left( \begin{array}{r} \frac{\sqrt{2}} {2} \\ \frac{\sqrt{2}} {2} \\ \end{array} \right) , \quad q_1 = \frac{\sqrt{2}} {2} \left( \begin{array}{r} -1 \\ 1 \\ \end{array} \right) = \left( \begin{array}{r} - \frac{\sqrt{2}} {2} \\ \frac{\sqrt{2}} {2} \\ \end{array} \right) . \end{equation*}

TRUE/FALSE: These vectors are mutually orthonormal.
Write the vector \(\left( \begin{array}{c} 4 \\ 2 \end{array} \right) \) as a linear combination of vectors \(q_0 \) and \(q_1 \text{.}\)