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The transpose (indicated by T) of any row vector is a column vector, and the transpose of any column vector is a row vector: […] = [] and [] = […]. The set of all row vectors with n entries in a given field (such as the real numbers ) forms an n -dimensional vector space ; similarly, the set of all column vectors with m entries forms an m ...
The row space of this matrix is the vector space spanned by the row vectors. The column vectors of a matrix. The column space of this matrix is the vector space spanned by the column vectors. In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column ...
While the terms allude to the rows and columns of a two-dimensional array, i.e. a matrix, the orders can be generalized to arrays of any dimension by noting that the terms row-major and column-major are equivalent to lexicographic and colexicographic orders, respectively. It is also worth noting that matrices, being commonly represented as ...
Formally, a multivariate random variable is a column vector = (, …,) (or its transpose, which is a row vector) whose components are random variables on the probability space (,,), where is the sample space, is the sigma-algebra (the collection of all events), and is the probability measure (a function returning each event's probability).
In Matlab/GNU Octave a matrix A can be vectorized by A(:). GNU Octave also allows vectorization and half-vectorization with vec(A) and vech(A) respectively. Julia has the vec(A) function as well. In Python NumPy arrays implement the flatten method, [note 1] while in R the desired effect can be achieved via the c() or as.vector() functions.
When a vector has only zeros and ones as entries, it is called a logical vector, a special case of a logical matrix. The logical operation and takes the place of multiplication. The outer product of two logical vectors ( u i ) and ( v j ) is given by the logical matrix ( a i j ) = ( u i ∧ v j ) {\displaystyle \left(a_{ij}\right)=\left(u_{i ...
In the simple case where we consider the vector space , a ket can be identified with a column vector, and a bra as a row vector. If, moreover, we use the standard Hermitian inner product on C n {\displaystyle \mathbb {C} ^{n}} , the bra corresponding to a ket, in particular a bra m | and a ket | m with the same label are conjugate transpose .
A circulant matrix is fully specified by one vector, , which appears as the first column (or row) of . The remaining columns (and rows, resp.) of C {\displaystyle C} are each cyclic permutations of the vector c {\displaystyle c} with offset equal to the column (or row, resp.) index, if lines are indexed from 0 {\displaystyle 0} to n − 1 ...