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Column vector

 

 
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Column vector
 
 
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In linear algebra
Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
, a column vector or column matrix is an m × 1 matrix
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
, i.e. a matrix consisting of a single column of elements.

The transpose
Transpose

In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
 of a column vector is a row vector
Row vector

In linear algebra, a row vector or row matrix is a 1 × n matrix , that is, a matrix consisting of a single row:The transpose of a row vector is a column vector:...
 and vice versa.

The set of all column vectors forms a vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
 which is the dual space
Dual space

In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra....
 to the set of all row vectors.

implify writing column vectors in-line with other text, sometimes they are written as row vectors with the transpose operation applied to them.

or


For further simplification, some authors also use the convention of writing both column vectors and row vectors as rows, but separating row vector elements with commas
Comma (punctuation)

The comma is a punctuation mark. It has the same shape as an apostrophe or single closing quotation mark in many typefaces, but it differs from them in being placed on the baseline of the text....
 and column vector elements with semicolons (see alternative notation 2 in the table below).






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Encyclopedia


In linear algebra
Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
, a column vector or column matrix is an m × 1 matrix
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
, i.e. a matrix consisting of a single column of elements.

The transpose
Transpose

In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
 of a column vector is a row vector
Row vector

In linear algebra, a row vector or row matrix is a 1 × n matrix , that is, a matrix consisting of a single row:The transpose of a row vector is a column vector:...
 and vice versa.

The set of all column vectors forms a vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
 which is the dual space
Dual space

In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra....
 to the set of all row vectors.

Notation

To simplify writing column vectors in-line with other text, sometimes they are written as row vectors with the transpose operation applied to them.

or


For further simplification, some authors also use the convention of writing both column vectors and row vectors as rows, but separating row vector elements with commas
Comma (punctuation)

The comma is a punctuation mark. It has the same shape as an apostrophe or single closing quotation mark in many typefaces, but it differs from them in being placed on the baseline of the text....
 and column vector elements with semicolons (see alternative notation 2 in the table below). This alternative notation is used, for instance, in MATLAB
MATLAB

MATLAB is a Numerical analysis environment and programming language. Maintained by The MathWorks, MATLAB allows easy matrix manipulation, plotting of function and data, implementation of algorithms, creation of user interfaces, and interfacing with programs in other languages....
, a widely used programming language
Programming language

A programming language is a machine-readable artificial language designed to express computations that can be performed by a machine, particularly a computer....
 specifically designed to simplify computations involving matrices.

Row vector Column vector
Standard matrix notation
Alternative notation 1
Alternative notation 2


Operations

  • Matrix multiplication
    Matrix multiplication

    In mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix. This article gives an overview of the various ways to perform matrix multiplication....
     involves the action of multiplying each row vector of one matrix by each column vector of another matrix.


  • The dot product
    Dot product

    In mathematics, the dot product, also known as the scalar product, is an operation which takes two vector over the real numbers R and returns a real-valued scalar quantity....
     of two vectors a and b is equivalent to multiplying the row vector representation of a by the column vector representation of b:


See also

Covariance and contravariance of vectors