Calculation Results
Dimension (Rank):
Basis for Column Space:
';
for (var j = 0; j < cols; j++) {
// Default values for a simple example (Rank 2)
var val = 0;
if (i === j) val = 1;
if (j === 3) val = i + 1; // 4th column linear combo
html += '';
}
html += '
';
}
container.innerHTML = html;
document.getElementById('result-output').style.display = 'none';
document.getElementById('error-message').style.display = 'none';
}
function calculateMatrixBasis() {
var rows = parseInt(document.getElementById('rows-input').value);
var cols = parseInt(document.getElementById('cols-input').value);
var matrix = [];
var originalMatrix = [];
// 1. Read Matrix Data
for (var i = 0; i < rows; i++) {
var row = [];
var origRow = [];
for (var j = 0; j < cols; j++) {
var cellId = 'cell-' + i + '-' + j;
var val = parseFloat(document.getElementById(cellId).value);
if (isNaN(val)) {
document.getElementById('error-message').innerText = "Invalid input at Row " + (i+1) + ", Col " + (j+1);
document.getElementById('error-message').style.display = 'block';
return;
}
row.push(val);
origRow.push(val);
}
matrix.push(row);
originalMatrix.push(origRow);
}
// 2. Gaussian Elimination to find Pivot Columns (RREF)
var pivotCols = [];
var lead = 0;
for (var r = 0; r < rows; r++) {
if (cols <= lead) break;
var i = r;
// Find pivot
while (Math.abs(matrix[i][lead]) < 1e-10) {
i++;
if (i === rows) {
i = r;
lead++;
if (cols === lead) break;
}
}
if (cols <= lead) break; // Should be covered above, but safe check
// Swap rows
var temp = matrix[i];
matrix[i] = matrix[r];
matrix[r] = temp;
// Store pivot column index
pivotCols.push(lead);
// Normalize row
var val = matrix[r][lead];
for (var j = 0; j < cols; j++) {
matrix[r][j] /= val;
}
// Eliminate other rows
for (var i = 0; i < rows; i++) {
if (i === r) continue;
val = matrix[i][lead];
for (var j = 0; j < cols; j++) {
matrix[i][j] -= val * matrix[r][j];
}
}
lead++;
}
// 3. Display Results
var resultContainer = document.getElementById('basis-vectors');
resultContainer.innerHTML = '';
if (pivotCols.length === 0) {
document.getElementById('rank-result').innerText = "0 (Zero Matrix)";
resultContainer.innerHTML = "No basis (only the zero vector)";
} else {
document.getElementById('rank-result').innerText = pivotCols.length;
// Extract basis vectors from ORIGINAL matrix based on pivot indices
for (var k = 0; k < pivotCols.length; k++) {
var colIndex = pivotCols[k];
var vectorHtml = '';
vectorHtml += 'v' + (k+1) + ' = [';
var components = [];
for (var r = 0; r < rows; r++) {
// Format output to avoid ugly floats like 3.0000000004
var val = originalMatrix[r][colIndex];
var displayVal = Math.abs(Math.round(val) – val) < 1e-10 ? Math.round(val) : val.toFixed(4);
components.push(displayVal);
}
vectorHtml += components.join(', ') + ']T
';
resultContainer.innerHTML += vectorHtml;
}
}
document.getElementById('result-output').style.display = 'block';
document.getElementById('error-message').style.display = 'none';
}
Understanding Matrix Basis and Linear Algebra
In linear algebra, determining the basis of a matrix is a fundamental operation used to understand the structure of the vector space spanned by the matrix's columns (the Column Space). This Matrix Basis Calculator allows you to input vectors in matrix form and automatically identifies the linearly independent vectors that form a basis for that space.
What is a Basis?
A basis for a vector space is a set of vectors that satisfies two critical conditions:
- Linear Independence: No vector in the set can be written as a linear combination of the others.
- Spanning: Every vector in the space can be written as a linear combination of the basis vectors.
When dealing with a matrix, the "Column Space" is the set of all possible linear combinations of its column vectors. Often, a matrix contains redundant information—vectors that are just combinations of other vectors. Finding a basis helps us simplify the dataset to its essential components.
How the Calculator Works
This tool utilizes Gaussian Elimination to convert the input matrix into Reduced Row Echelon Form (RREF). Here is the logic behind the calculation:
- Input: You define an $m \times n$ matrix where each column represents a vector in $\mathbb{R}^m$.
- Reduction: The calculator performs row operations to identify "pivot" positions. A pivot is the first non-zero entry in a row after reduction.
- Identification: The columns in the Reduced Row Echelon form that contain pivots correspond to the linearly independent columns in the original matrix.
- Result: The calculator extracts these specific columns from your original input. These form the Basis for the Column Space.
Rank and Dimension
The calculator also outputs the Rank of the matrix. The rank is simply the number of vectors in the basis. It tells you the dimension of the column space.
- If Rank = Number of Columns, the vectors are all linearly independent.
- If Rank < Number of Columns, the vectors are linearly dependent (some are redundant).
Example Calculation
Consider a matrix with columns $v_1 = [1, 2, 3]^T$, $v_2 = [2, 4, 6]^T$, and $v_3 = [0, 1, 1]^T$.
Notice that $v_2$ is exactly $2 \times v_1$. Therefore, $v_2$ is redundant. The calculator will identify that the pivot columns are 1 and 3. It will return $\{v_1, v_3\}$ as the basis, and calculate a Rank of 2.
Applications
Finding a basis is essential in fields such as:
- Data Compression: Reducing dimensions by keeping only essential features (Principal Component Analysis).
- Solving Systems of Equations: Determining if a solution exists and if it is unique.
- Physics and Engineering: Analyzing forces and states in mechanical systems where redundant constraints exist.