Basis and Dimension Calculator

Linear Algebra Concepts:

\[ \text{Dimension} = \text{rank}; \quad \text{Basis} = \text{linearly independent vectors spanning the space} \]

Dimension:

Basis Vectors:

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1. What Are Basis and Dimension?

In linear algebra, the dimension of a vector space is the number of vectors in any basis for the space. A basis is a set of linearly independent vectors that span the full space.

2. How the Calculator Works

The calculator performs these steps:

\[ \text{1. Construct matrix from input vectors} \] \[ \text{2. Reduce to row echelon form} \] \[ \text{3. Count pivots for dimension} \] \[ \text{4. Extract basis vectors} \]

Key Concepts:

Linear Independence: No vector can be written as a combination of others
Spanning: All vectors in the space can be formed from basis combinations
Rank: Number of linearly independent rows/columns in a matrix

3. Importance in Linear Algebra

Applications: Basis and dimension are fundamental for solving systems of equations, understanding transformations, and working with vector spaces in physics and engineering.

4. Using the Calculator

Instructions: Enter vectors as comma-separated components, with different vectors separated by semicolons. Example: "1,0,0; 0,1,0; 0,0,1" for standard basis in ℝ³.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dimension and rank?
A: For a matrix, the rank equals the dimension of its column space (or row space).

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

Q3: What's the standard basis?
A: In ℝⁿ, vectors with 1 in one position and 0 elsewhere (e.g., (1,0), (0,1) in ℝ²).

Q4: How does dimension relate to coordinates?
A: The dimension determines how many coordinates are needed to specify any vector in the space.

Q5: What if my vectors are dependent?
A: The calculator will eliminate dependent vectors to find a basis with dimension ≤ number of input vectors.