The set of harmonic functions on a given open set " U " can be seen as the kernel of the Laplace operator ? and is therefore a vector space over "'R "': sums, differences and scalar multiples of harmonic functions are again harmonic.
32.
Recall that " n "-dimensional projective space \ mathbb { P } ^ n is defined to be the set of equivalence classes of non-zero points in \ mathbb { A } ^ { n + 1 } by identifying two points that differ by a scalar multiple in " k ".
33.
The set of real scalar multiples of this null vector, called a " null line " through the origin, represents a " line of sight " from an observer at a particular place and time ( an arbitrary event we can identify with the origin of Minkowski spacetime ) to various distant objects, such as stars.
34.
When the condition number is exactly one ( which can only happen if " A " is a scalar multiple of a linear isometry ), then a solution algorithm can find ( in principle, meaning if the algorithm introduces no errors of its own ) an approximation of the solution whose precision is no worse than that of the data.
35.
This holds more generally for any algebra " R " over an uncountable algebraically closed field " k " and for any simple module " M " that is at most countably-dimensional : the only linear transformations of " M " that commute with all transformations coming from " R " are scalar multiples of the identity.
36.
Alternatively, it is possible to rewrite the equation of the plane using dot products with \ mathbf { p } in place of the original dot product with \ mathbf { v } ( because these two vectors are scalar multiples of each other ) after which the fact that \ mathbf { p } is the closest point becomes an immediate consequence of the Cauchy Schwarz inequality.
37.
Scalar multiples of the identity operator of course commute with everything, and are usually identified with the scalars themselves, so one can seemingly add a scalar to an operator without multiplying it with I first . ( The self-adjoint operators are, however, closed under the operation p, q \ mapsto i ( pq-qp ), which makes them into a Lie algebra.
38.
The elements of the polynomial ring k [ x _ 0, \ dots, x _ n ] are not functions on \ mathbb { P } ^ n because any point has many representatives that yield different values in a polynomial; however, for homogeneous polynomials the condition of having zero or nonzero value on any given projective point is well-defined since the scalar multiple factors out of the polynomial.
39.
If a vector x maximizes R ( M, x ), then any non-zero scalar multiple kx also maximizes R, so the problem can be reduced to the Lagrange problem of maximizing \ sum _ { i = 1 } ^ n \ alpha _ i ^ 2 \ lambda _ i under the constraint that \ sum _ { i = 1 } ^ n \ alpha _ i ^ 2 = 1.
40.
Specializing further, if it happens that " M " has a Riemannian metric on which " G " acts transitively by isometries, and the stabilizer subgroup " G " x of a point acts irreducibly on the tangent space of " M " at " x ", then the Casimir invariant of ? is a scalar multiple of the Laplacian operator coming from the metric.
