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!