For a given set of vectors S = {v1, v2,..., vn} 

express one vector as a linear combination of the others.

Another wayto say this is to state that there are no solutions for α1, α2, and α3 in thehomogeneous equation

other than the trivial solution α1 = α2 = α3 = 0.

It is important to realize that the concepts of linear independence and dependence are defined only for sets—

Diagonal Dominance.

For . A matrix An×n

that all diagonally dominant matrices are nonsingular.

Vandermonde Matrices

the columns in V constitute a linearly independent set whenever n ≤ m.

The Lagrange interpolation polynomial 

of degree m − 1.

when t=x1, l(t)=y1

Maximal Independent Subsets

Basic Facts of Independence

The Wronski 28 matrix

If there is at least one point x = x0 such that W(x0) is nonsingular, that S must be a linearly independent set.

Prove is really simple: nonsingular = independent from the assumpiton

s