1. What is a Basis in Linear Algebra?

A basis is a set of linearly independent vectors that span a vector space. It provides a coordinate system for the vector space where every vector can be uniquely expressed as a linear combination of the basis vectors.

2. How Does Basis Calculation Work?

The calculator finds a basis by identifying linearly independent vectors from the input set using methods like Gaussian elimination or row reduction.

Where:

• \( v_1, v_2, ..., v_n \) — Linearly independent vectors
• No vector can be written as a linear combination of the others
• The set spans the vector space

Explanation: The basis calculation involves determining which vectors are essential and which can be expressed as combinations of others.

3. Importance of Basis Vectors

Details: Basis vectors are fundamental in linear algebra for understanding vector spaces, solving systems of equations, and performing coordinate transformations.

4. Using the Calculator

Tips: Enter vectors as comma-separated coordinates, separated by semicolons. Example: "1,2,3;4,5,6" for two 3D vectors.

5. Frequently Asked Questions (FAQ)

Q1: What makes vectors linearly independent?
A: Vectors are linearly independent if no vector can be expressed as a linear combination of the others. The only solution to c₁v₁ + c₂v₂ + ... + cₙvₙ = 0 is c₁ = c₂ = ... = cₙ = 0.

Q2: Can there be multiple bases for the same vector space?
A: Yes, a vector space can have infinitely many different bases, but all bases for the same space have the same number of vectors (the dimension).

Q3: What is the standard basis?
A: The standard basis consists of vectors with 1 in one coordinate and 0 in all others. For R³, it's (1,0,0), (0,1,0), (0,0,1).

Q4: How is basis related to dimension?
A: The number of vectors in a basis equals the dimension of the vector space. A 3D space has a basis with 3 vectors.

Q5: What if my vectors are linearly dependent?
A: The calculator will remove dependent vectors and return only the linearly independent ones that form the basis.