Which of the following sets of vectors are linearly independent?

Question

A. [0 0], [-6 9]

B. [7 -4], [6 -9], [-8 -2]

C. [-6 8 7], [-4 6 -9], [10 -14 2]

D. [9 6 2], [0 1 -8], [3 4 -7]

E. [4 9 0], [-7 -5 0], [-3 2 0]

F. [3 -1 -2], [-1 5 -8]

Answer 1

1. Two of the sets of vectors are linearly dependent just by observing them: sets B and E. Basically, for B we have three vectors in a plane ( two coordinates). One of the vectors can be expressed as linear combination of the other two. The situation for E is the same, the third coordinate for the three vectors is 0, so the vectors are in xy plane, one of them can be expressed in terms of the other two.
2. Set A is linearly independent by observation.
3. F is linearly independent set.
4. In case of C and D, we need to row reduce the matrix of the vectors to see if we can have row of zeros.

[-6 8 7]

[-4 6 -9]

[10 -14 2]

We can add first and second row and will get

[(-6 -4) (8 + 6) (7 - 9)] = [ -10 14 - 2] = - [ 10 -14 2]

We see, in case of C, the third row is linear combination of the first and second row. This means C is linearly dependent.

You need to perform similar check for D to see if it is dependent.

(independent)