Subspaces Worksheet with Answer Key

What's on the worksheet

1. Decide whether each set is a subspace.
   \{(x, y, z) : x = 3z\} in \mathbb{R}^3\{(x, y) : x \leq y\} in \mathbb{R}^2\{(x, y, z) : z = xy\} in \mathbb{R}^3\{(x, y, z) : x = 0 \text{ or } y = 0\} in \mathbb{R}^3\{(x, y, z) : 2x - y = 0 \text{ and } z = 5\} in \mathbb{R}^3
2. A 2 × 2 table of numbers with rows (p, q) and (r, s) is recorded as the vector (p, q, r, s) in R^4.
3. Let A = \begin{bmatrix} 1 & 2 & -2 & -2 & -3 \\ -1 & -1 & 4 & 2 & 5 \\ -2 & -3 & 6 & 5 & 10 \end{bmatrix}. Write the solution set N of Ax = 0 as a…
4. In R^4, let V = span\(1, 0, 1, 2), (0, 1, -1, 1)\ and let H be the set of vectors with x_1 + 2x_2 - x_3 + x_4 = 0.
5. For real numbers s and t, let W_1 be the set of all vectors (2s - t + 1,\ s + t + 2,\ 3s + 3,\ -t) and W_2 the set of all vectors (2s - t + 3,\ s +…
6. Let U and W be subspaces of R^n, and let U + W be the set of all sums u + w with u U and w W.