What are Linearly Dependent Vectors?
Two vectors are defined as linearly dependent if at least one of the vectors in the set is a linear combination of the other vectors. A linear combination is an expression of the sum of two vectors multiplied by a constant. The concepts of linear dependence and independence are central to the understanding of vector space.
How to Determine Linearly Dependent Vectors
The vectors u=<2,-1,1>, v=<3,-4,-2>, and w=<5,-10,-8> are dependent since the determinant is zero.
To find the relation between u, v, and w we look for constants x, y, and z such that
This is a homogeneous system of equations. Using Gaussian Elimination, we see that the matrix
in row-reduced form is:
Thus, y=-3z and 2x=-3y-5z=-3(-3z)-5z=4z which implies 0=xu+yv+zw=2zu-3zv+zw or equivalently w=-2u+3v. A quick arithmetic check verifies that the vector w is indeed equal to -2u+3v.