Question

Find the dependency of vector (1,0,3),(0,1,2),(2,3,1),(4,1,0)

Answer 1

Let x,w,y,z represent the vectors in that order. We have to determine whether we can find real numbers r,s,t,u such that rx + sw + ty + uz = 0

To find r,s,t,u we solve the matrix equation

1 0 2 4| 0

0 1 3 1| 0

3 2 1 0| 0

Taking R3;= R3 - 3R1 we obtain the matrix below

1 0 2 4.| 0

0 1 3 1| 0

0 2 -5 -12|0

Taking again R3;= R3 - 2R2,we obtain the matrix below

1 0 2 4| 0

0 1 3 1|0

0 0 -11 -4|0

The set of equations has non zero solutions. Therefore {x,w,y,z} is a linearly dependent set of vectors.