Test whether the columns of a matrix are linearly independent

Returns TRUE when the QR-decomposition rank of V equals the number of columns, signalling that the columns of V are linearly independent and so define a non-degenerate simplicial cone.

Usage

is_linearly_independent(V, tol = 1e-10)

Arguments

V

A numeric matrix whose columns are vectors in \(\mathbb{R}^n\).

tol

Numeric. Reserved for future tolerance control; the current implementation delegates the rank test to qr, whose default tolerance is used.

Value

A single logical: TRUE if ncol(V) columns are linearly independent, FALSE otherwise.

Details

Linear independence of the generators is a precondition for every solid-angle backend in the package. The dispatcher compute_solid_angle issues a warning when this test fails because the spanned cone is then lower-dimensional and has solid-angle measure zero.

References

Strang, G. (2016). Introduction to Linear Algebra, 5th edition. Wellesley-Cambridge Press. ISBN 978-0980232776.

See also

compute_solid_angle, diagnose_cone for downstream consumers; is_positive_definite, is_tridiagonal for sibling tests.

Examples

is_linearly_independent(cbind(c(1, 0), c(0, 1)))   # TRUE
#> [1] TRUE
is_linearly_independent(cbind(c(1, 0), c(2, 0)))   # FALSE
#> [1] FALSE