Calculator
Formula Used
Vectors v1, v2, ..., vk are linearly dependent when scalars
a1, a2, ..., ak, not all zero, satisfy:
a1v1 + a2v2 + ... + akvk = 0
The rank test is:
rank([v1 v2 ... vk]) = k means independent.
If rank([v1 v2 ... vk]) < k, the vectors are dependent.
The nullity is k - rank.
How To Use This Calculator
- Enter one vector per line, or choose column entry.
- Separate coordinates with commas or spaces.
- Set tolerance for rounding or measured values.
- Choose decimal places for the final display.
- Press Calculate to view the result below the header.
- Use CSV or PDF download for saving the report.
Example Data Table
| Example | Vectors | Rank | Decision | Reason |
|---|---|---|---|---|
| A | [1, 2, 3], [2, 4, 6], [1, 0, 1] | 2 | Dependent | Second vector is twice the first vector. |
| B | [1, 0, 0], [0, 1, 0], [0, 0, 1] | 3 | Independent | Every vector adds a new direction. |
| C | [1, 1], [2, 3], [4, 5] | 2 | Dependent | Three vectors in two dimensions must depend. |
Vector Linear Dependence Guide
Linear dependence shows whether a vector set carries repeated direction information. A set is dependent when one vector can be built from the others. A set is independent when no vector is redundant. This calculator checks that idea with rank, pivots, row reduction, and the null space. It works for two dimensional, three dimensional, and larger vectors.
Why Dependence Matters
Dependence is important in algebra, geometry, data science, mechanics, graphics, and signal work. Independent vectors can form a basis when they also span a space. Dependent vectors cannot give unique coordinates. They add columns or rows, but they do not add a new direction. That makes rank smaller than the number of vectors.
What The Tool Checks
The calculator reads each vector from your input. It builds a dependence matrix, reduces it to row echelon form, and finds the rank. It also finds pivot vectors and free variables. When free variables exist, the tool reports a nonzero relation. That relation proves dependence, because its coefficients combine the vectors into the zero vector.
Reading The Result
If rank equals the number of vectors, the set is linearly independent. If rank is lower, the set is linearly dependent. The nullity tells how many independent relations exist. A nullity of zero means no nonzero relation exists. A positive nullity means at least one vector is unnecessary for spanning the same space.
Good Input Habits
Use one vector per line when vectors are entered as rows. Separate coordinates with commas or spaces. Keep every vector the same length. Fractions such as 3/4 are accepted. Use a smaller tolerance for exact classroom data. Use a larger tolerance when measurements contain rounding noise.
Practical Uses
Students can verify homework steps. Teachers can prepare examples. Engineers can inspect direction sets. Analysts can test feature vectors before modeling. The downloadable report helps keep results, formulas, and reduced matrices together. The example table also shows how changing one row can create or remove redundancy.
Accuracy Notes
Floating point results can hide tiny errors. The tolerance setting treats very small values as zero. This helps noisy data, but exact symbolic work may need stricter settings. Always compare rank, nullity, pivots, and relations together for confirmation.
FAQs
What does linearly dependent mean?
It means at least one vector can be written as a combination of the others. The set has redundant direction information.
What does linearly independent mean?
It means no vector can be built from the remaining vectors. Every vector adds a new direction to the set.
How does the calculator test dependence?
It builds a matrix from the vectors. Then it finds rank using row reduction. Rank is compared with the number of vectors.
What is rank?
Rank is the number of pivot directions in a matrix. It shows how many vectors are truly independent.
What is nullity?
Nullity is the number of free coefficient variables. A positive nullity proves that a nonzero dependence relation exists.
Can I enter fractions?
Yes. You can enter values such as 1/2, -3/4, decimals, or whole numbers. Keep every vector the same length.
Why is tolerance needed?
Tolerance treats tiny values as zero. It helps when decimal input or measured data creates small rounding errors.
When is the determinant useful?
The determinant helps when the dependence matrix is square. A nonzero determinant means the square vector set is independent.