Scalar Product and Norms
Scalar productThe scalar product (or, dot product) between two vectors The scalar product is sometimes denoted We say that the vectors are orthogonal if Matlab syntax
>> x = [1; 2; 3]; y = [4; 5; 6]; >> scal_prod = x'*y; Examples: Vector normsMeasuring the size of a scalar value is unambiguous — we just take the magnitude (absolute value) of the number. However, when we deal with higher dimensions, and try to define the notion of size, or length, of a vector, we are faced with many possible choices. Norms are real-valued functions that satisfy a basic set of rules that a sensible notion of size should involve. You can consult the formal definition of a norm here. In this course, we focus on the following three popular norms for a vector
Matlab syntax
>> x = [1; 2; -3]; >> r2 = norm(x,2); % l2-norm >> r1 = norm(x,1); % l1 norm >> rinf = norm(x,inf); % l-infty norm Examples:
Cauchy-Schwartz inequality, anglesThe Cauchy-Schwartz inequality allows to bound the scalar product of two vectors in terms of their Euclidean norm. Cauchy-Schwartz inequality:
For any two vectors When none of the vectors The Cauchy-Schwartz inequality can be generalized to other norms, using the concept of dual norm. Example: Orthonormal basisA basis Example: An orthonormal basis in Hyperplanes and half-spacesHyperplanesA hyperplane is a set described by a single affine equality. Precisely, an hyperplane in Hyperplanes are affine sets, of dimension Example: Half-spacesA half-space is a subset of Based on the notion of angle between two vectors, we can understand the meaning of an inequality of the form |