Question

# let v = the x1x3 plane (also called the xz plane) in r3, and v1 = (1, 0, 2), v2 = (0, 0, 1). is v contained in span{v1, v2}?

Answer

v is not contained in span{v1, v2}.

- Q:What is v? A: v = {(x1,0,x3) | x1,x3 ∈ R}
- Q:How to check if v is contained in span{v1, v2}? A:v is contained in span{v1, v2} if and only if there exist scalars c1, c2 such that v = c1v1 + c2v2.
- Q:Solve for c1 and c2. A:v = c1v1 + c2v2, so (x1,0,x3) = c1(1, 0, 2) + c2(0, 0, 1)
- Q:What is the augmented matrix corresponding to this system? A:The augmented matrix is [1 0 | x1; 0 0 | x3].
- Q:Solve the augmented matrix using row reduction. A:[1 0 | x1; 0 0 | x3] -> [1 0 | x1; 0 0 | x3-x1/2].
- Q:What are the possible solutions? A:The system has a unique solution if and only if x3=x1/2. In this case, c1 = x1 and c2 = 0. If x3 is not equal to x1/2, then the system has no solution, and v is not contained in span{v1, v2}.