Solve the problem. Let v1 = -1 V2 = 1 V3 = -5, and H = Span -2 -2 {vL_ v2 v3} . Note that v3 = 3 v 1 - 2 v 2. Which of the following sets form a basis for the subspace H, i.e., which sets form an efficient spanning set containing no unnecessary vectors? A: {v1, v2 v3} B: {v1, v2} C: {v1, v3} D: {v2, v3} %3D B and C A only B only B, C, and D

Suppose that V is a that has subspaces of U and W. Furthermore, suppose that {u¹, u2} is a basis for U, that {1, 2} is a basis for W and that the only vector that U and W have in common is the zero vector 0. Show that {u¹, u², w¹, w²}. After you get done, note where you used the fact that the only vector common to both U and W is 0. (Without this condition the vectors {u¹, u², w², w2} don't need to be linearly independent, so if you didn't use this condition you definitely made a mistake somewhere.)

Find a basis for the solution space of the given homogeneous linear system. X4- 2x2 + 16x3 = 0 2x, 3x2 + 23x3 = 0 A basis for the solution space of the given homogeneous linear system is { }. (Use a comma to separate vectors as needed.)

Let u1, U2, ..., un be a set of vectors, and suppose there are nontrivial solutions to x11x2u2+ +xnun = 0 Explain why, if x is a free variable, you can always express u; in terms of the vectors corresponding to the basic variables.