Question

Determine which of these sets spans R^3

q) (1,0,1) , (3,1,0), (-1,0,0),(2,1,5)

x) (2,1,2) , (1,1,1), (-3,0,-3)

y) (-1,2,1),(4,1,-3),(-6,3,5)

z) (1,0,0),(0,2,0),(1,2,0),(0,-1,1)

a) only q

b)q and x

c) q and z

d) only z

Answer 1

we just need to verify that the linear combinations of the vectors should span whole of r^3

In other words the system Ax = b should have solutions for all values of b, we know this possible if A has non zero determinant;

q) for this sytem if any 3 vectors of the 4 vectors have a non-zero determinant then it spans r^3.

taking the first three , the determinant is 1 so no further calculation. This systems spans R^3.

x) is the only possible matrix, this has determinant of zero, this can easily be seen as

3*(r1-r2) = r3 .

y) has a determinant of zero (cant be easily verified without straightforward calculation). Hence cant span r^3

z) I will consider the last three vectors (as the first three dont have a third component). .

This has a determinant of -2. So this spans r^3.

so the option is C. q and z only span r^3.