Hi everyone,

I have the following question that needs confirmation and clarification. This is a review for my elementary differential equations course.

Question: Which of the following sets of vectors are linearly independent?

A. [ 18 ] , [ -9 ]

[ -8 ] , [ 4 ]

B. [ 5 ] , [ -6 ] , [ -1 ]

[ -9 ] , [ -5 ] , [ -14 ]

[ -3 ] , [ 7 ] , [ 4 ]

C. [ 6 ] , [ 3 ] , [ 2 ]

[ -7 ] , [ 8 ] , [ 4 ]

[ 0 ] , [ 0 ] , [ 0 ]

D. [ 1 ] , [ 6 ]

[ 3 ] , [ 1 ]

E. [ 8 ]

[ 2 ]

F. [ 0 ] , [ 5 ]

[ 0 ] , [ -7 ]

Attempt At solving:

A. Since the second matrix is a scalar multiple of the first matrix, the must be linearly dependent so that one is out.

B. If you add together matrix one and matrix 2 together, the result is the 3rd vector which is a linear combination of other vectors in the set so therefore must be linearly dependent as well.

C. None of the vectors in this set are deemed a scalar multiple or a linear combination so must be said as linearly independent.

D. Same as the C.

E. I'm not entirely sure on this one, but would it require at least 2 vectors to be considered as linearly independent?

F. F should not be because there is not a leading 1 so it cannot be linearly independent.

I would like to know which ones I have chosen are correct and which reasoning I have stated above are wrong as this is a review for my differential equations course. I did not previously learnt his material in linear algebra so i am not entirely sure!