Linear Independence

In linear algebra, a set of vectors is said to be linearly independent if none of the vectors in the set can be expressed as a linear combination of the others. In other words, a set of vectors {v1, v2, ..., vn} is linearly independent if the only way to express the zero vector 0 as a linear combination of these vectors is with all coefficients equal to zero.

Mathematically, this can be expressed as:

a1v1 + a2v2 + ... + an*vn = 0

where ai are scalars, implies that a1 = a2 = ... = an = 0.

Example:

Consider the vectors {[1, 2], [3, 4]} in R². To show these vectors are linearly independent, we must demonstrate that the only way to express the zero vector [0, 0] as a linear combination of them is with all coefficients equal to zero.

Let a and b be scalars such that:

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

This gives us the system of equations:

a + 3b = 0
2a + 4b = 0

Solving this system, we find that a = b = 0. Therefore, these two vectors are linearly independent.

Note that if we had a third vector [5, -6], then the set {[1, 2], [3, 4], [5, -6]} would not be linearly independent, since the first two vectors could be expressed as a linear combination of the last two.