1. What is a Basis?

A basis is a set of linearly independent vectors that span a vector space. For any given set of vectors, we can find a basis by eliminating linearly dependent vectors through row reduction (Gaussian elimination).

2. How Basis Calculation Works

The calculator performs the following steps:

1. Forms a matrix with the input vectors as rows
2. Performs Gaussian elimination to reduce the matrix to row echelon form
3. Identifies the pivot rows, which correspond to the basis vectors
4. Returns the original vectors that correspond to these pivot rows

3. Importance of Basis

Details: Finding a basis is fundamental in linear algebra as it provides the minimal set of vectors needed to represent any vector in the space. It's used in solving systems of equations, transformations, and many other applications.

4. Using the Calculator

Tips: Enter one vector per line, with components separated by commas. All vectors must have the same dimension. The calculator will return a maximal linearly independent subset (a basis) for the space spanned by your vectors.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between a basis and a spanning set?
A: A basis is a spanning set with no redundant vectors - all vectors in a basis are linearly independent.

Q2: Can there be more than one basis for a space?
A: Yes, any vector space (except {0}) has infinitely many bases, but all bases for a space have the same number of vectors (the dimension).

Q3: What if my vectors are linearly dependent?
A: The calculator will return only the linearly independent vectors that form a basis for the space spanned by your input.

Q4: How does row reduction help find a basis?
A: Row reduction preserves the row space (span of the vectors) while making linear dependencies obvious through zero rows.

Q5: What's the maximum number of basis vectors I can get?
A: For vectors in ℝⁿ, the maximum basis size is n (the dimension of the space).