Homework 9 - QR factorizations and linear transformations

1. Find the QR decomposition of \(A = \begin{pmatrix} 1 & -1 & 1 & -1 \\ -1 & 0 & 1 & -1 \\ 0 & 1 & -1 & 1 \\ -1 & 1 & 0 & 1 \end{pmatrix}\)

2. Assume we have a QR factorization \(A = \begin{pmatrix} \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{6}} & \frac{-2}{\sqrt{6}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 \\ \end{pmatrix} \begin{pmatrix} 1 & \frac{2}{3} & 0 \\ 0 & 1 & \frac{1}{2} \\ 0 & 0 & 1 \end{pmatrix}\) Without resorting to Sage, compute the inverse of $A$. (Hint: these particular type of matrices have simple formulas for inverses.)

3. Assume that $T: V \to W$ is a bijective linear transformation of vector spaces. For each $w \in W$ pick some $v \in V$ with $T(v) = w$ and define \(\begin{aligned} T^{-1} : W & \to V \\ w & \mapsto v \end{aligned}\) Show that this is a well-defined function and is a linear transformation.

4. Again assume that $T$ is a linear transformation and a bijection and $T^{-1}$ is as in problem 3. Show that \(T(T^{-1}(w)) = w\) for all $w \in W$ and \(T^{-1}(T(v)) = v\) for all $v \in V$. This says that $T^{-1}$ is the inverse to $T$ under composition. Thus being a bijective linear transformation is equivalent to have an inverse (on both sides) which is a linear transformation.

5. Given two vector spaces $V$ and $W$ show that $\operatorname{Map}(V,W)$, the set of linear transformations from $V$ to $W$, has the structure of a vector space.

   For a $v \in V$, show the function \(\begin{aligned} ev_v : \operatorname{Map}(V,W) & \to W \\ T & \mapsto T(v) \end{aligned}\) is a linear transformation.