In linear algebra, a set of vectors is considered linearly dependent if at least one vector in the set can be defined as a linear combination of the others. Essentially, this means there is "redundancy" in the set; you don't need all the vectors to describe the space they occupy because one or more can be built using the remaining vectors.
To understand how this differs from sets where every vector provides unique information, you may want to read our guide on Linearly Independent Vectors. For a side-by-side comparison of these two fundamental states, see our overview of Linear Dependence and Independence.
The Formal Definition
A set of vectors {v1, v2, ..., vn} is linearly dependent if there exist scalars (constants) c1, c2, ..., cn, not all of which are zero, such that:
c1v1 + c2v2 + ... + cnvn = 0
If the only way to satisfy this equation is for every single constant (c) to be zero, the vectors are linearly independent.
How to Identify Linear Dependence
There are several ways to determine if a set of vectors is dependent:
- Scalar Multiples: If you have only two vectors, they are dependent if one is simply a multiple of the other (e.g., v1 = [1, 2] and v2 = [2, 4]).
- The Zero Vector: If a set of vectors contains the zero vector [0, 0, 0], the set is always linearly dependent.
- The Determinant Test: For a set of n vectors in n-dimensional space, you can place them into a matrix. If the determinant is zero, the vectors are linearly dependent.
- Visual Geometry: * In 2D, dependent vectors are collinear (they lie on the same line).
- In 3D, three dependent vectors are coplanar (they lie on the same flat plane).
If you find that your vectors do not meet these criteria and instead provide entirely unique directions, they are likely linearly independent vectors.
Practice Question & Solutions
Determine if the vectors u = (3, -6) and v = (-1, 2) are linearly dependent.
Check if one is a multiple of the other. u = (3, -6)
If we multiply v by -3: -3 * (-1, 2) = (3, -6).
Since u = -3v, the vectors are proportional and therefore linearly dependent.
Determine whether the vectors a = (1, 2, 3), b = (4, 5, 6), and c = (7, 8, 9) are linearly dependent.
Place the vectors into a 3x3 matrix and find the determinant: | 1 2 3 | | 4 5 6 | | 7 8 9 |
Calculation: 1(59 - 68) - 2(49 - 67) + 3(48 - 57) 1(45 - 48) - 2(36 - 42) + 3(32 - 35) 1(-3) - 2(-6) + 3(-3) -3 + 12 - 9 = 0
Since the determinant is 0, the vectors are linearly dependent.
Find the value of k that makes the following set of vectors linearly dependent: u = (1, k) v = (4, 8)
For two vectors to be dependent, their components must be proportional:
Multiply both sides by 8:
k = 2
If k = 2, then the vectors are linearly dependent.
Show that the set of vectors v1 = (1, 0, 0), v2 = (0, 1, 0), and v3 = (1, 1, 0) is linearly dependent.
We look for a linear combination: v3 = (1, 1, 0)
v1 + v2 = (1, 0, 0) + (0, 1, 0) = (1, 1, 0)
Because v3 = v1 + v2, the third vector is simply the sum of the first two. This satisfies the definition of linear dependence.
Check the linear dependence of: v1 = (2, 1) v2 = (0, 0)
The set contains the zero vector v2 = (0, 0). Any set containing a zero vector is linearly dependent because you can multiply the zero vector by any non-zero scalar to satisfy the equation c1v1 + c2v2 = 0 (e.g., 0v1 + 5v2 = 0).
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