Basis and Dimension Worksheet with Answer Key

What's on the worksheet

1. Find the dimension of each subspace and give one basis of it.
2. Let V = span\b_1, b_2\ in R^4, where b_1 = (1, -3, 2, -3) and b_2 = (1, -2, -1, -3).
3. Let v_1 = (1, -1, 2, 0), v_2 = (2, 1, 1, 3), v_3 = (3, 0, 3, 3), v_4 = (0, 3, -3, 3), v_5 = (0, 1, 0, 2), and let V = span\v_1, , v_5\ in R^4.
4. Let N be the set of solutions in R^5 of the system \begin{aligned} x_1 + x_2 - 2x_3 + 3x_5 &= 0 \\ x_1 + 2x_2 - 3x_3 + 2x_4 + x_5 &= 0.
5. Decide whether each statement is true or false.
6. Let v_1, v_2, v_3 be a basis of a subspace V of R^n, and let \mathbf{w}_1 = \mathbf{v}_1, \qquad \mathbf{w}_2 = \mathbf{v}_1 + \mathbf{v}_2, \qquad…