Question

how do we know that the vectors are linearly independent or dependent?

Answer 1

where are the vectors?

Answer 2

lets take ..
[0 5 3 2]
[-4 1 1 2]
[-2 3 2 3]

Answer 3

A set of vectors in R^n is linearly dependent if one of the vectors can be written as a linear combination of the rest.

Take the first two vectors. They are linearly independent b/c they are not scalar multiples of each other. You can tell by the 0 in the first slot of the first vector (no number times 0 can make -4).

Now take all three vectors. If this set were linearly dependent,
a[0 5 3 2] + b[-4 1 1 2] = [-2 3 2 3]
for some numbers a and b.

Because of the 0 in the first slot of the first vector, you know b=1/2. Then,
a[0 5 3 2] + (1/2)[-4 1 1 2] = [-2 3 2 3]
a[0 5 3 2] + [-2 1/2 1/2 1] = [-2 3 2 3]
a[0 5 3 2] = [0 5/2 3/2 2]

No number a can satisfy the last equation, so the three vectors must be linearly independent. These questions are usually easy if there are 0's involved.

Answer 4

hmm this one is better

Answer 5

okay.