Basis and Dimension Finder
Find the Basis Vectors and Dimension of the Column Space.
Dimension
Basis for Column Space
*These are columns from the original matrix corresponding to the pivots.
Column Space Visualization
Visual: Green columns are “Linearly Independent”. Gray columns are combinations of the green ones.
What is a Basis?
A Basis is the smallest set of vectors you need to “build” an entire space. Think of it like the primary colors (Red, Green, Blue).
Key Properties
- Span: You can make ANY vector in the space by mixing (adding/scaling) the basis vectors.
- Independence: No basis vector is redundant. You can’t make “Red” by mixing Green and Blue.
Dimension
The Dimension is simply the number of vectors in the basis.
- A Line has Dimension 1 (1 basis vector).
- A Plane has Dimension 2 (2 basis vectors).
- Our physical space is Dimension 3.
How to Find the Basis
1. Put the matrix into Reduced Row Echelon Form (RREF).
2. Identify the columns that have Leading 1s (Pivots).
3. The Basis consists of the Original Columns (from the starting matrix) that correspond to those pivot positions.
Note: Do not use the columns from the RREF matrix itself as the basis for the Column Space; use the original ones!
Frequently Asked Questions (FAQ)
Q: Is the basis unique?
A: No! Just like you could pick Cyan, Magenta, and Yellow to mix colors, there are infinitely many valid bases for a space. However, the number of vectors (Dimension) is always the same.
Q: Can the zero vector be in a basis?
A: Never. The zero vector adds no new direction or span. It is linearly dependent on everything.
Q: What if the matrix has no pivots?
A: This means it’s the Zero Matrix. The dimension is 0, and the basis is the empty set.