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A scalar is an element of a field which is used to define a vector space.In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector.
For example, while a given person has a specific age, height and weight, the representation of these features of an unspecified person from within a group would be a random vector. Normally each element of a random vector is a real number .
The space of vectors may be considered a coordinate space where elements are associated with a list of elements from K. The units of the field form a group K × and the scalar-vector multiplication is a group action on the coordinate space by K ×. The zero of the field acts on the coordinate space to collapse it to the zero vector.
The matrix has the same properties as , therefore we can choose as an element of the kernel with at least one positive entry. Hence c = sup i ∈ [ | 1 , n | ] z i p i {\displaystyle c=\sup _{i\in [|1,n|]}{\frac {z_{i}}{p_{i}}}} is nonnegative and reached with at least one value k ∈ [ | 1 , n | ] {\displaystyle k\in [|1,n|]} .
Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.
In linear algebra, a column vector with elements is an matrix [1] consisting of a single column of entries, for example, = [].. Similarly, a row vector is a matrix for some , consisting of a single row of entries, = […]. (Throughout this article, boldface is used for both row and column vectors.)
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