| 1 | 2 | 3 |
| 2 | 4 | 6 |
| 1 | 1 | 0 |
| Rank | 2 |
|---|---|
| Pivot Columns | 1, 2 |
| Basis Vectors | [1,2,1], [2,4,1] |
| Interpretation | Column 3 is dependent on columns 1 and 2. |
- Column space: Col(A) = span{a₁, a₂, …, aₙ}, where aⱼ are columns of A.
- Basis from pivots: Columns of the original A corresponding to pivot columns in RREF(A) form a basis for Col(A).
- Rank: rank(A) = number of pivot columns = dimension of Col(A).
- Orthonormal basis: Apply Gram–Schmidt to basis vectors to get orthonormal q-vectors.
- Projection: p = QQᵀb, where Q contains orthonormal basis vectors as columns.
- Choose Paste or Grid input mode.
- Enter your matrix A; fractions like 5/2 are allowed.
- Pick output decimals and, if needed, adjust the numerical tolerance.
- Optional: enter a vector b to compute its projection onto Col(A).
- Press Submit; results appear above the form.
- Use the download buttons to export CSV or PDF reports.
Column space as an output set
For an m×n matrix A, each column is a vector in R^m. The column space is the set of all linear combinations Ax, so it describes every output vector you can produce by varying x. If m=4 and n=6, outputs still live in R^4, but only within a subspace of dimension r=rank(A). This calculator reports r, pivot indices (1-based), and a spanning basis directly from your input.
Pivot columns and basis selection
Row reduction does not change dependence relations among columns. After computing RREF(A), the columns that contain leading 1s identify pivot positions. Selecting those same columns back in the original matrix gives a basis for Col(A) with r vectors. For example, if pivots occur in columns 1 and 3, then {a1,a3} spans the same space as all n columns, while non‑pivot columns are exact linear combinations. The reported basis is minimal, so you never keep redundant vectors.
Rank as the dimension indicator
Rank provides a quick capacity metric: dim(Col(A))=rank(A)=r. With n columns, the remaining n−r degrees of freedom correspond to relationships among columns and to the null space of Aᵀ. If r=0, every entry is numerically zero and Col(A) contains only the zero vector. If r=m, the columns span all of R^m and every b has an exact solution Ax=b (when n≥m and A is full row rank). This is useful for diagnosing underdetermined versus overdetermined models.
Orthonormal basis and projection data
When you enable the orthonormal option, the basis vectors are converted into q-vectors with unit length and mutual orthogonality using Gram–Schmidt. Orthonormal bases make projections stable: p=QQᵀb is the closest vector to b inside Col(A) in least‑squares distance. The coefficient vector c=Qᵀb reports how much of b lies along each q direction, and the residual b−p quantifies the part of b that cannot be explained by the columns. Monitoring ‖b−p‖ helps compare competing feature sets.
Tolerance, precision, and export usage
Numerical data often arrives with rounding, so the tolerance setting decides when a small value counts as zero while forming pivots. A default around 1e−10 works for clean integers, while noisy decimals may need 1e−8 or 1e−6. The CSV export lists pivots, rank, and basis vectors for quick checking in a spreadsheet. The PDF export packages the same results for homework solutions, lecture notes, or audit trails.
What are pivot columns, and why do they define the basis?
Pivot columns are the columns of RREF(A) that contain leading 1s. Those indices mark independent directions. Taking the same columns from the original matrix gives r independent vectors that span Col(A).
Does row-reducing change the column space?
Row operations change the actual columns, but they preserve which original columns are dependent or independent. That is why the calculator uses RREF only to detect pivots, then returns the pivot columns from A as the basis.
How should I choose the tolerance value?
Use about 1e−10 for exact integers or clean decimals. If your entries come from measurements or rounding, increase to 1e−8 or 1e−6 so tiny numerical noise is treated as zero during pivot detection.
Why is my rank smaller than the number of columns?
Rank r counts independent columns. If r < n, some columns can be written as linear combinations of others, so they do not add new directions to the span. The basis output keeps only the essential columns.
What does the projection p = QQᵀb mean?
It is the closest vector to b that lies in the column space, minimizing ‖b−p‖. If b is already in Col(A), then p equals b. Otherwise, the residual b−p is the part not explained by A.
Can I enter fractions and negative numbers?
Yes. Use formats like 5/2, −3, or −0.75 in either Paste or Grid mode. Fractions are converted to decimals internally, and results are shown with your selected precision.