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Basis Calculation:
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A basis is a set of linearly independent vectors that span a vector space. It provides the fundamental building blocks for representing any vector in that space through linear combinations.
The calculator finds a basis by:
Details: Basis calculation is fundamental for understanding vector spaces, solving systems of linear equations, and performing coordinate transformations.
Tips: Enter vectors as comma-separated coordinate values. Each vector should have the same dimension for proper basis calculation.
Q1: What makes a set of vectors a basis?
A: A set of vectors forms a basis if they are linearly independent and span the entire vector space.
Q2: Can different bases exist for the same vector space?
A: Yes, a vector space can have infinitely many different bases, but all bases 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 (e.g., (1,0,0), (0,1,0), (0,0,1) in R³).
Q4: How is basis related to dimension?
A: The number of vectors in any basis of a vector space equals the dimension of that space.
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.