The basic definition:

A set of two or more vectors is linearly dependent whenever at least one of them is a linear combination of the others, and linearly independent if none of them is a linear combination of the others.

Another way to say it:

A set of two or more vectors is linearly dependent whenever at least one of them lies in the span of the others, and linearly independent if none of them lies in the span of the others.

Example. In 2-space, the vectors i and j are linearly independent (or more accurately, the set {i, j} is linearly independent) since neither i nor j can be written as a linear combination of the other. Similarly, in 3-space, the vectors i, j and k are linearly independent, but the vectors i, j, k and i + k are linearly dependent, since the last is the sum of the first and third.

Example. Are the vectors [2, 3], [3, 4] and [1, 1] in 2-space linearly independent or linearly dependent?

Check if any of them is a linear combination of the others, i.e. check whether any of the following equations has a solution:

[2, 3] = a[3, 4] + b[1, 1]

[3, 4] = a[2, 3] + b[1, 1]

[1, 1] = a[2, 3] + b[3, 4]

It's easy to see that the first equation has the solution a = 1, b = -1, so

[2, 3] = (1)[3, 4] + (-1)[1, 1] .

The three vectors are thus linearly dependent.

Example. Are the vectors [1, 2, 3], [4, 5, 6] and [1, 0, 1] in 3-space linearly independent or linearly dependent?

Check if any of them is a linear combination of the others, i.e. check if any of the following equations has a solution.

[1, 2, 3] = a[4, 5, 6] + b[1, 0, 1]

[4, 5, 6] = a[1, 2, 3] + b[1, 0, 1]

[1, 0, 1] = a[1, 2, 3] + b[4, 5, 6].

Each equation is equivalent to a linear system of three equations in two variables. All three systems turn out to have no solution, i.e. none of the three vector equations has a solution, so the vectors are linearly independent.

Example. Suppose we have a set of vectors {u, v, w, ..., 0} in either 2-space or 3-space. This set must be linearly dependent, since

0 = 0u + 0v + 0w + ... .

In general, any set of vectors containing the zero vector must be linearly dependent.