Basis and Dimension Calculator

Check whether two vectors form a basis for R² and compute the span dimension.

Linear AlgebraVectorsCollege Math

Calculator

v₁ x
v₁ y
v₂ x
v₂ y
Please enter valid values.

What this calculator teaches

A basis is a set of independent vectors that spans a space.

In R², two vectors form a basis exactly when the 2×2 determinant is nonzero.

Formula & Symbols

ConceptFormulaMeaning
Basis for R²det([v1 v2]) ≠ 0The vectors are independent and span the plane.
Dimensionnumber of independent directionsDimension of the span.

Step-by-Step Examples

Example 1
v₁=(1,2), v₂=(3,4)
  • det=1·4-2·3=-2.
  • The determinant is not zero.
✓ The vectors form a basis for R².

Where students use this

🎓
College mathematics
Use the tool to connect notation, formulas, and numeric answers.
🧪
Engineering analysis
Apply the result to models, systems, approximations, and design calculations.
💻
Computer science
Use matrix, vector, numerical, and optimization ideas in algorithms and data work.

Common Mistakes to Avoid

⚠️
Ignoring restrictions

Check denominator, domain, endpoint, and matrix-invertibility conditions before trusting a result.

⚠️
Rounding too early

Carry extra decimals through intermediate steps and round only the final answer.

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Mixing symbolic and numeric answers

Some topics need exact symbolic reasoning, while this calculator gives a clear numerical learning check.

Frequently Asked Questions

Is this calculator meant for students? ▾
Yes. It gives an answer and shows the main method so students can learn the reasoning instead of copying only the result.
Can I use decimals and negative numbers? ▾
Yes, when the topic allows them. Some formulas have restrictions such as positive radii, nonzero denominators, or valid intervals.
Why can textbook answers look different? ▾
Equivalent algebraic forms, different rounding, or using exact notation instead of decimals can make answers look different while meaning the same thing.

Related Math Calculators

🔢 Determinant Calculator🔢 Null Space and Column Space Calculator🔢 Vector Calculator

Formula Explorer connections

Interpretation: This relationship represents vectors, linear systems or transformations using matrix structure and vector operations. Assumption: Matrix dimensions must be compatible. Inversion, decomposition and unique solutions require rank, conditioning or nonzero-determinant conditions.

Cross Product Calculator →Determinant Calculator →Dot Product Calculator →Math Formula Explorer →