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

###### Homework4.6.1.1.

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 $\min_x \| b - A x \|_2$

###### Homework4.6.1.2.

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 $q_1 \text{.}$