1. What is a Basis in Vector Space?

A basis is a set of linearly independent vectors that span a vector space. Any vector in the space can be uniquely expressed as a linear combination of the basis vectors.

2. How Basis Calculation Works

The calculator uses Gaussian elimination to find a basis from the given set of vectors by identifying linearly independent vectors that span the same space.

3. Importance of Basis Vectors

Details: Basis vectors provide a coordinate system for the vector space and are fundamental in linear algebra for solving systems of equations, transformations, and more.

4. Using the Calculator

Tips: Enter vectors separated by semicolons, with components separated by commas. All vectors must have the same dimension.

5. Frequently Asked Questions (FAQ)

Q1: What makes a set of vectors a basis?
A: A set of vectors is a basis if they are linearly independent and span the entire vector space.

Q2: Can a vector space have multiple bases?
A: Yes, any vector space has infinitely many bases, but all bases have the same number of vectors (dimension).

Q3: What is the standard basis?
A: The standard basis consists of vectors with one component equal to 1 and all others 0 (e.g., (1,0,0), (0,1,0), (0,0,1) in R³).

Q4: How is linear independence determined?
A: Vectors are linearly independent if the only solution to their linear combination equaling zero is all coefficients being zero.

Q5: What is the dimension of a vector space?
A: The dimension is the number of vectors in any basis for that space.