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 Gaussian elimination on the input vectors to:

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
• Dimension: Uniquely determined for any finite-dimensional space

3. Importance in Linear Algebra

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

4. Using the Calculator

Instructions: Enter vectors separated by semicolons, with components separated by commas. 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 basis and spanning set?
A: A basis must be linearly independent, while a spanning set may have redundant vectors.

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

Q3: What's the standard basis for ℝⁿ?
A: The set of vectors with 1 in one component and 0 elsewhere (e.g., (1,0), (0,1) in ℝ²).

Q4: How does dimension relate to rank?
A: The rank of a matrix equals the dimension of its column space.

Q5: What about infinite-dimensional spaces?
A: This calculator handles finite dimensions only (e.g., function spaces require advanced methods).