Practical Linear Algebra, 3rd Edition

B

Selected Exercise Solutions

Chapter 1

• 1a. The triangle vertex with coordinates (0.1, 0.1) in the [d1, d2]-system is mapped to

  $\begin{array}{l}{x}_1=0.9\times 1+0.1\times 3=1.2,\\ x_2=0.9\times 2+0.1\times 3=\mathrm{2.1.}\end{array}$

  The triangle vertex (0.9, 0.2) in the [d1, d2]-system is mapped to

  $\begin{array}{l}{x}_1=0.1\times 1+0.9\times 3=2.8,\\ x_2=0.8\times 2+0.2\times 3=\mathrm{2.2.}\end{array}$

  The triangle vertex (0.4, 0.7) in the [d1, d2]-system is mapped to

  $\begin{array}{l}{x}_1=0.6\times 1+0.4\times 3=1.8,\\ x_2=0.3\times 2+0.7\times 3=\mathrm{2.7.}\end{array}$

• 1b. The coordinates (2, 2) in the [e1, e2]-system are mapped to

  $\begin{array}{l}{u}_1=\frac{2-1}{3-1}=\frac{1}{2},\\ u_2=\frac{2-2}{3-2}=0.\end{array}$

• 3. The local coordinates (0.5, 0, 0.7) are mapped to

  $\begin{array}{l}{x}_1=1+0.5\times 1=1.5,\\ x_2=1+0\times 2=1,\\ x_3=1+0.7\times 4=\mathrm{3.8.}\end{array}$

  4. We have simply moved Exercise 1a to 3D, so the first two coordinates are identical. Since the 3D triangle local coordinates are at the extents of the local frame, they will be at the extents ...