Linear Independence Calculator - Determinant Test for Vectors

The linear independence calculator tells you in one step whether a set of vectors is linearly independent or linearly dependent. You pick a 2D or 3D vector space, enter up to three vectors as coordinates, and the tool computes the determinant of the column matrix or the cross product magnitude for two 3D vectors, then prints a clear verdict.

• • Checking homework: Quickly verify whether the answer to a textbook problem treats the given set as independent or dependent.
• • Building a basis: Confirm a candidate set of vectors forms a valid basis for R^2 or R^3 before you build change-of-basis matrices.
• • Solving linear systems: Check that coefficient columns are independent before assuming a unique solution to a system of equations.
• • Testing subspaces: Decide whether a small set of vectors really spans the space you are working in or collapses onto a lower-dimension subspace.

Use the linear independence calculator any time a problem says 'are these vectors linearly independent?' and you want a check that is faster and less error-prone than hand expansion. The tool is built for 2D and 3D vectors, which cover most introductory linear algebra assignments.

When you also need the length of each vector, Vector Magnitude Calculator gives you the Euclidean norm using the same coordinate inputs.

The tool stacks the entered vectors as the columns of a square matrix, then applies the determinant test that mathematicians use to classify a set as linearly independent or dependent.

• v1, v2, v3: Vectors you enter, given as components (x, y) for 2D or (x, y, z) for 3D.
• det: Determinant of the matrix whose columns are the entered vectors.
• |v1 x v2|: Magnitude of the cross product, used when two 3D vectors are tested.
• Tolerance: A small threshold of 1e-10 used to treat near-zero results as exactly zero for the verdict.

The linear independence calculator returns the actual numeric value of the determinant so you can also use the result as a sanity check when you do the expansion by hand.

When you choose two 3D vectors, the tool switches from the determinant to the cross product magnitude. This avoids the need for a third vector and keeps the same independent-versus-dependent classification rule: non-zero means independent, zero means dependent.

According to Wolfram MathWorld, a set of vectors is linearly independent if and only if the only linear combination that equals the zero vector is the trivial one with all coefficients equal to zero.

When the vectors come from the columns of a coefficient matrix, System of Equations Calculator will solve the matching system so you can connect independence to unique solutions.

These four ideas are the backbone of the linear independence test and they show up the moment you start working with vector spaces.

These concepts are also the building blocks of related tools. The Matrix Calculator, for example, computes the determinant of any 2x2 or 3x3 matrix, which is the exact value the linear independence test relies on.

If you want to see the matrix determinant expanded by hand, Matrix Calculator performs the same calculation with extra matrix operations like transpose and inverse.

Follow these four steps to classify any small set of vectors as independent or dependent.

1. 1 Pick a dimension: Use the Dimension selector to choose 2D for plane vectors or 3D for vectors in space.
2. 2 Choose the number of vectors: Pick 2 vectors for a quick pairwise check, or 3 vectors when the problem gives a triple to test.
3. 3 Enter the components: Type each component into the matching x, y, or z box. Components default to the standard basis vectors so the page opens to a known independent set.
4. 4 Read the verdict: The Results panel shows the determinant, the cross product magnitude when it applies, and a clear Linearly Independent or Linearly Dependent label.

Try v1 = (2, 1) and v2 = (4, 2). The calculator will show det = 2*2 - 1*4 = 0 and mark the set as linearly dependent, because v2 is exactly twice v1.

If your vectors turn out to be independent, the column matrix has a non-zero determinant and is therefore invertible, and Adjoint Matrix Calculator builds the adjugate and inverse from the same coordinate matrix in one step.

These benefits come straight from how students, teachers, and engineers use the tool in day-to-day linear algebra work.

• • Real-time verdict: You get a clear Linearly Independent or Linearly Dependent label in real time as you type, with no submit step required.
• • No manual sign tracking: The calculator handles the cofactor expansion and the cross product formula, so you do not have to remember which signs to flip in a 3x3 determinant.
• • Cross product fallback: Two 3D vectors get a cross product magnitude instead of a determinant, which avoids the ambiguity of leaving a slot empty for a third vector.
• • Working shown on screen: The determinant and the cross product magnitude both display on the page, so you can cite the actual numeric value in homework or reports.
• • Edge case coverage: Zero vectors, scalar multiples, and three 2D vectors are all handled with a clear dependent verdict and a short reason.

The tool is also useful as a teaching aid because it shows the numeric value of the determinant alongside the verdict, so learners can compare the two and build intuition for why the rule works.

If you are deciding which vectors to keep in a basis, Cross Product Calculator can quickly show the area spanned by two candidates so you can pick a well-conditioned set.

Several inputs and assumptions can flip the verdict, so it helps to know which knobs change the answer and which leave it alone.

• • Only 2D and 3D vector spaces are supported. For higher dimensions, you would need a 4x4 or larger determinant that the calculator does not compute.
• • Coefficients are real numbers. The tool does not handle complex entries, polynomial entries, or symbolic coefficients.
• • The verdict uses a strict zero test, so very nearly dependent sets (for example, a 0.0001 scalar multiple) may be classified by their actual numeric determinant rather than by an exact symbolic zero.

These caveats matter most when the vectors come from measurement data, where small rounding errors can make an independent set look dependent. For symbolic textbook problems the rules above are usually all you need.

According to Wolfram MathWorld, a set of n vectors in R^n is linearly independent exactly when the determinant of the matrix with those vectors as columns is non-zero.

If a near-zero determinant is making the verdict unstable, Dot Product Calculator can report the pairwise angle between candidates so you can see whether two vectors are nearly collinear before trusting the verdict.