Then e . g . \ vec { 0 } \ in C _ W and, and the vector addition, the scalar multiplication, and the inner product is declared as, and, respectively.
42.
Scalar multiplication is the multiplication of a vector by a scalar ( where the product is a vector ), and must be distinguished from inner product of two vectors ( where the product is a scalar ).
43.
Conversely, one can prove all orthogonal polarities are induced in this way, and that two symmetric bilinear forms with trivial radical induce the same polarity if and only if they are equal up to scalar multiplication.
44.
However, the class L ^ + is in general not closed under subtraction and scalar multiplication by negative numbers; one needs to further extend it by defining a wider class of functions L with these properties.
45.
A consequence of this relation is that also \ mathbf { C } can be seen as an element of a projective space; two camera matrices are equivalent if they are equal up to a scalar multiplication.
46.
Under this identification, addition and scalar multiplication of vectors in " V " correspond to addition and scalar multiplication of their coordinate vectors in "'F " "'n ".
47.
Under this identification, addition and scalar multiplication of vectors in " V " correspond to addition and scalar multiplication of their coordinate vectors in "'F " "'n ".
48.
Note that using Shamir's trick, a sum of two scalar multiplications u _ 1 \ times G + u _ 2 \ times Q _ A can be calculated faster than two scalar multiplications done independently.
49.
Note that using Shamir's trick, a sum of two scalar multiplications u _ 1 \ times G + u _ 2 \ times Q _ A can be calculated faster than two scalar multiplications done independently.
50.
The elements of " L " are the " vectors " and the elements of " K " are the " scalars ", with vector addition and scalar multiplication obtained from the corresponding field operations.
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