Orthogonal Bases and Projection Onto a Subspace Worksheet with Answer Key

What's on the worksheet

1. Let u_1 = (1, a, 2), u_2 = (1, 1, b) and u_3 = (1, -1, 0).
2. Let W = span\u_1, u_2\ in R³, with u_1 = (1, 1, 0) and u_2 = (1, -1, 1), and let y = (-2, -2, 6).
3. Let u_1 = (1, 2, 1, 0), u_2 = (2, -1, 0, 1), u_3 = (1, 0, -1, -2), u_4 = (0, 1, -2, 1) and y = (-1, -7, 3, 7).
4. Let W be the column space of the matrix U = \frac{1}{2}\begin{bmatrix} 1 & 1 \\ 1 & -1 \\ 1 & 1 \\ 1 & -1 \end{bmatrix}.
5. Let W be the set of solutions in R^4 of the system \begin{aligned} x_1 + x_2 + x_3 + x_4 &= 0 \\ x_1 - x_2 + 2x_3 &= 0, \end{aligned} and let r_1 =…