Home
Back
Matrix Rank and Image Space Basis:
| From: | To: |
The basis of the image space (or column space) of a matrix consists of linearly independent vectors that span the space of all possible outputs of the linear transformation represented by the matrix. The dimension of this space is called the rank of the matrix.
The calculator computes the rank and basis using matrix operations:
The basis vectors are typically found by performing Gaussian elimination on the matrix and identifying the pivot columns.
Details: The rank of a matrix reveals the dimension of its image space, which is crucial for understanding the properties of linear transformations, solving systems of linear equations, and analyzing vector spaces.
Tips: Enter the matrix row by row, with elements separated by spaces or commas. The matrix should represent a valid linear transformation.
Q1: What is the relationship between rank and image space?
A: The rank of a matrix equals the dimension of its image space (column space).
Q2: How is the basis of image space determined?
A: The basis consists of the linearly independent columns of the matrix, typically identified through row reduction.
Q3: What does a full rank matrix indicate?
A: A matrix has full rank if its rank equals the number of columns, meaning all columns are linearly independent.
Q4: Can the rank exceed the matrix dimensions?
A: No, the rank cannot exceed the number of rows or columns of the matrix.
Q5: How does rank relate to invertibility?
A: A square matrix is invertible if and only if it has full rank.