Linear Independence Calculator

Check if a set of vectors is linearly independent or dependent using determinant analysis.

Enter Vectors

Vector v₁

Vector v₂

Result

Status
--
Determinant --
Number of Vectors --
Dimension --
Conclusion --

Step-by-Step Solution

det(M) != 0 => Linearly Independent

Understanding Linear Independence

A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. In other words, the only solution to the equation c₁v₁ + c₂v₂ + ... + cₙvₙ = 0 is the trivial solution where all scalars c₁, c₂, ..., cₙ are zero.

If a non-trivial solution exists (at least one scalar is non-zero), the vectors are linearly dependent, meaning one or more vectors can be expressed as combinations of the others.

Testing for Linear Independence

Determinant Method

Form a matrix with the vectors as columns. If the determinant is non-zero, the vectors are independent.

det(M) != 0 => Independent

2x2 Determinant

For two 2D vectors forming matrix [[a,b],[c,d]].

det = ad - bc

3x3 Determinant

For three 3D vectors, use cofactor expansion along the first row.

det = a(ei-fh) - b(di-fg) + c(dh-eg)

Why Linear Independence Matters

Linear independence is a core concept in linear algebra with far-reaching implications:

  • Basis of a Vector Space: A basis is a set of linearly independent vectors that spans the entire vector space. The number of vectors in any basis is the dimension of the space.
  • Solving Linear Systems: A system of linear equations has a unique solution if and only if the coefficient vectors are linearly independent.
  • Matrix Invertibility: A square matrix is invertible if and only if its column vectors are linearly independent.
  • Rank: The rank of a matrix equals the maximum number of linearly independent column (or row) vectors.

Geometric Interpretation

In 2D, two vectors are linearly independent if they do not lie on the same line through the origin. They span the entire plane. In 3D, three vectors are linearly independent if they do not all lie in the same plane through the origin. They span all of 3D space.

Common Misconceptions

  • Linear independence does not mean the vectors are perpendicular (orthogonal). Orthogonal vectors are always independent, but independent vectors need not be orthogonal.
  • If one vector is the zero vector, the set is automatically linearly dependent.
  • More vectors than the dimension of the space are always linearly dependent (e.g., 3 vectors in 2D).