A collection of vectors is said to be *linearly dependent *if some non-trivial linear combination of them equals zero.

More formally, vectors v_{1},v_{2},…,v_{n} are said to be linearly dependent if there exist scalars a_{1},a_{2},…,a_{n} with at least one a_{i} nonzero such that,

a_{1}v_{1}+a_{2}v_{2}+…+a_{n}v_{n} = 0.

The vectors are said to be linearly independent if they are *not linearly dependent*. This means that:

If a_{1}v_{1}+a_{2}v_{2}+…+a_{n}v_{n} = 0 then a_{1}=a_{2}=…=a_{n}=0.

**How to Check Linear Dependence:**

- We arrange the
*components of the vectors*as rows of a matrix. - Calculate the determinant of the resulting matrix to check whether it is nonzero.
- If the determinant is zero the vectors are dependent.
- If the determinant is nonzero the vectors are independent.

**Example 1:**

Consider the three vectors as shown below.

a = (1, 0, 2).

b = (3, 4, -1).

c = (4, 4, 1).

Taking the components of the three vectors to be rows of a matrix we obtain the following matrix,

We now calculate the determinant of the above matrix.

det(A) = 1*(4 + 4) – 0*(3 + 4) + 2*(12 – 16) = 8 – 0 + (-8) = 0.

Since the determinant of the above matrix is zero we conclude that the three given vectors are linearly dependent.

This can also be seen very easily as follows,

(4, 4, 1) = (1, 0, 2) + (3, 4, -1).

c = a + b.

a + b – c = 0.

Since a non-trivial linear combination of the vectors is nonzero, this means that the vectors are linearly dependent.

This gives us another way of verifying linear dependence.

**Example 2:**

Consider the three vectors as shown below.

a = (3, 1, 0).

b = (0, -1, 2).

c = (4, 1, 2).

We now calculate the determinant of the above matrix,

det(A) = 3*(-2 – 2) – 1*(0 – 8) + 0*(0 + 4) = -12 + 8 + 0 = -4.

Since the determinant is nonzero we conclude that the three vectors are linearly independent.