There are many situations when we might wish to know whether a set of vectors is linearly dependent, that is if one of the vectors is some combination of the others.

Two vectors **u** and **v** are linearly independent if the only
numbers x and
y satisfying x**u**+y**v**=0 are x=y=0. If we let

then x**u**+y**v**=0 is equivalent to

If **u** and **v** are linearly independent, then the only
solution to this system of equations is the trivial solution, x=y=0. For
homogeneous systems this
happens precisely when the determinant
is non-zero. We have now found a test for determining whether a given
set of vectors is linearly independent: A set of n vectors of length n is
linearly independent if the matrix with these vectors as columns has a non-zero
determinant. The set is of course dependent if the determinant is zero.

The vectors <1,2> and <-5,3> are linearly independent since the matrix

has a non-zero determinant.

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=x**u**+y**v**+z**w**=2z**u**-3z**v**+z**w**
or equivalently **w**=-2**u**+3**v**. A quick arithmetic check verifies
that the vector
**w** is indeed equal to -2**u**+3**v**.

**[Vector Calculus Home]
[Math
254 Home] [Math 255 Home]
[Notation]
[References]**

**Copyright **© **1996 Department
of Mathematics, Oregon State
University**

If you have questions or comments, don't hestitate to contact us.