Range space of a matrix of rationals:

Range space of a machine-precision matrix:

Range space of a complex matrix:

Range space of an arbitrary-precision matrix:

Range space of an exact matrix:

Find the range space symbolically:

The above result is generic; special values of the symbols may give different results:

Range space of a tall rectangular matrix:

Range space of a wide rectangular matrix:

The range space of a large numerical matrix is computed efficiently:

Range space of a sparse matrix:

Range space of a structured matrix:

Compute the range space for HilbertMatrix:

Range space of a matrix with finite field elements:

Compute the range space of a 10×12 matrix of univariate polynomials of degree 100:

m is a 3×3 random matrix of integers between 0 and 4:

The range space is full dimensional in ordinary arithmetic:

Use arithmetic modulo 5 to compute the range space:

Create a matrix with a linear dependency and noise added:

Compute the range space:

A tolerance setting commensurate with the noise eliminates a vector of noise-level values:

By default, symbolic expressions are considered nonzero:

In this case, RangeSpace is missing the special case that produces a singular matrix:

Write a function that tests if an expression might be zero:

Pass test[k] as the value of ZeroTest to get a more symbolic reduction:

This gives a valid, if nonstandard, basis for generic values of :

For , a two-dimensional range space is shown, even if the output includes an extraneous zero vector:

The extraneous vector can be removed, for example, with DeleteCases:

The following matrix has a one-dimensional range space:

Any vector of the form will lie along the line parallel to :

Generate 50 random points in the range space:

Visualize how the points lie along the line generated by :

The following matrix has a two-dimensional range space:

Any vector of the form will lie in the plane generated by and :

Generate 2500 random points in the range space:

Visualize how the points lie on the plane generated by and :

The following matrix has a three-dimensional range space:

Since is , the range space is all of , and has a solution for any value of :

Generate 2500 random points in the range space:

Visualize how the points fill a solid region:

The following system of equations has no solution:

Rewrite the system in matrix form:

The range space of the coefficient matrix is a plane:

The vector does not lie in this plane, so the system has no solutions:

The following overdetermined system of equations has a solution:

Rewrite the system in matrix form:

The range space of the coefficient matrix is a plane:

The vector lies in this plane, so the system has a solution:

Create a visualization of the fact that . Compute a basis for with RangeSpace:

Compute a basis for with NullSpace and Transpose:

Use InfiniteLine and InfinitePlane to visualize the two perpendicular subspaces:

Determine if the following system of equations has a unique solution:

Rewrite the system in matrix form:

The range space is full dimensional, so the system has a unique solution:

Verify the result using Solve:

Find all values of the for which the following overdetermined linear system has a solution:

Rewrite the system in matrix form:

Find the range space of the coefficient matrix :

Assign the to be the components of a general linear combination of basis vectors for the range space:

Use Solve to confirm that it is possible to solve for , and for every value of the :

Find all values of the vector for which the equation has a solution, and the general solution :

is a 3×3 singular matrix whose range space is not full dimensional:

for which there is a solution must be a linear combination of range space basis vectors:

Confirm that there is a solution for each such by finding one using LinearSolve:

All solutions are given by , where is any vector in the null space:

Construct the general solution:

Confirm the solution:

Determine if the following matrix has an inverse:

The range space is not full dimensional, so the matrix is not invertible:

Verify the result using Inverse:

Determine if the following matrix has a nonzero determinant:

Since the range space is full dimensional, the matrix must have nonzero determinant:

Confirm the result using Det:

Estimate the fraction of random 10×10 0–1 matrices that are invertible:

Find the dimension of the column space of the following matrix:

Column space is just another name for range space, so the answer can be computed directly:

Find the dimension of the subspace spanned by the following vectors:

By definition, this is the dimension of the range space of the matrix whose columns are the vectors:

The following three vectors are not linearly independent:

Find the range space of the matrix whose columns are the vectors:

The dimension of the range space is less than the number of vectors because of the linear dependence:

The following three vectors are linearly independent:

Find the range space of the matrix whose columns are the vectors:

The dimension of the range space equals the number of vectors because of the linear independence:

Determine if the following vectors are linearly independent or not:

The range space of is less than four dimensional, so they are not linearly independent: