Jump to content

Row and column vectors: Difference between revisions

From Wikipedia, the free encyclopedia
Content deleted Content added
AnomieBOT (talk | contribs)
m Dating maintenance tags: {{Fact}}
Tag: Reverted
→‎Matrix transformations: main tag, links, ce presentation
Tags: Reverted Mobile edit Mobile web edit Advanced mobile edit
(2 intermediate revisions by the same user not shown)
Line 91: Line 91:
\end{bmatrix} \,. </math>
\end{bmatrix} \,. </math>


==Preferred input vectors for matrix transformations==
==Matrix transformations==
{{main|Transformation matrix}}
{{POV|date=February 2021}}
An ''n'' × ''n'' matrix ''M'' can represent a [[linear map]] and act on row and column vectors as the linear map's [[transformation matrix]]. For a row vector ''v'', the product ''vM'' is another row vector ''p'':
Frequently a row vector presents itself for an operation within ''n''-space expressed by an ''n'' × ''n'' matrix ''M'',


:<math> v M = p \,.</math>
:<math> v M = p \,.</math>


Then ''p'' is also a row vector and may present to another ''n'' × ''n'' matrix ''Q'',
Another ''n'' × ''n'' matrix ''Q'' can act on ''p'',


:<math> p Q = t \,. </math>
:<math> p Q = t \,. </math>


Conveniently, one can write ''t'' = ''p Q'' = ''v MQ'' telling us that the [[matrix product]] transformation ''MQ'' can take ''v'' directly to ''t''. Continuing with row vectors, matrix transformations further reconfiguring ''n''-space can be applied to the right of previous outputs.
Then one can write ''t'' = ''p Q'' = ''v MQ'', so the [[matrix product]] transformation ''MQ'' maps ''v'' directly to ''t''. Continuing with row vectors, matrix transformations further reconfiguring ''n''-space can be applied to the right of previous outputs.


In contrast, when a column vector is transformed to become another column under an ''n'' × ''n'' matrix action, the operation occurs to the left,
When a column vector is transformed to another column vector under an ''n'' × ''n'' matrix action, the operation occurs to the left,


:<math> p^\mathrm{T} = M v^\mathrm{T} \,,\quad t^\mathrm{T} = Q p^\mathrm{T} </math>,
:<math> p^\mathrm{T} = M v^\mathrm{T} \,,\quad t^\mathrm{T} = Q p^\mathrm{T} </math>,


leading to the algebraic expression ''QM v''<sup>T</sup> for the composed output from ''v''<sup>T</sup> input. The matrix transformations mount up to the left in this use of a column vector for input to matrix transformation.
leading to the algebraic expression ''QM v''<sup>T</sup> for the composed output from ''v''<sup>T</sup> input. The matrix transformations mount up to the left in this use of a column vector for input to matrix transformation.

The column vector approach to matrix transformation leads to a [[right-to-left]] orientation for successive transformations. In [[geometric transformation]]s described by matrices, the two approaches are related by the [[transpose]] operator. Though equivalent, the fact that directionality of English text is [[left-to-right]] has led some English authors to have a preference for the row vector input to matrix transformation:

For instance, this row vector input convention has been used to good effect by Raiz Usmani,<ref>Raiz A. Usmani (1987) ''Applied Linear Algebra'' [[Marcel Dekker]] {{isbn|0824776224}}. See Chapter 4: "Linear Transformations"</ref> where on page 106 the convention allows the statement "The product mapping ''ST'' of ''U'' into ''W'' [is given] by:
:<math>\alpha (ST) = (\alpha S) T = \beta T = \gamma</math>."
(The Greek letters represent row vectors).

[[Ludwik Silberstein]] used row vectors for spacetime events; he applied Lorentz transformation matrices on the right in his [[List of important publications in physics#Special|Theory of Relativity]] in 1914 (see page 143).
In 1963 when [[McGraw-Hill]] published ''Differential Geometry'' by [[Heinrich Guggenheimer]] of the [[University of Minnesota]], he used the row vector convention in chapter 5, "Introduction to transformation groups" (eqs. 7a,9b and 12 to 15). When [[H. S. M. Coxeter]] reviewed<ref>Coxeter [http://www.ams.org/mathscinet/pdf/188842.pdf Review of ''Linear Geometry''] from [[Mathematical Reviews]]</ref> ''Linear Geometry'' by [[Rafael Artzy]], he wrote, "[Artzy] is to be congratulated on his choice of the 'left-to-right' convention, which enables him to regard a point as a row matrix instead of the clumsy column that many authors prefer." [[J. W. P. Hirschfeld]] used right multiplication of row vectors by matrices in his description of projectivities on the [[Galois geometry]] PG(1,q).<ref>[[J. W. P. Hirschfeld]] (1979) ''Projective Geometry over Finite Fields'', page 119, [[Clarendon Press]] {{isbn|0-19-853526-0}}</ref>

In the study of stochastic processes with a [[stochastic matrix]], it is conventional to use a row vector as the [[stochastic vector]].<ref>[[John G. Kemeny]] & [[J. Laurie Snell]] (1960) ''Finite Markov Chains'', page 33, D. Van Nostrand Company</ref>


== See also ==
== See also ==

Revision as of 18:18, 15 March 2021

In linear algebra, a column vector is a column of entries, for example,

Similarly, a row vector is a row of entries[1]

Throughout, boldface is used for both row and column vectors. The transpose (indicated by T) of a row vector is the column vector

and the transpose of a column vector is the row vector

The set of all row vectors with n entries forms an n-dimensional vector space; similarly, the set of all column vectors with m entries forms an m-dimensional vector space.

The space of row vectors with n entries can be regarded as the dual space of the space of column vectors with n entries, since any linear functional on the space of column vectors can be represented as the left-multiplication of a unique row vector.

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

Some authors also use the convention of writing both column vectors and row vectors as rows, but separating row vector elements with commas and column vector elements with semicolons (see alternative notation 2 in the table below).[citation needed]

Row vector Column vector
Standard matrix notation
(array spaces, no commas, transpose signs)
Alternative notation 1
(commas, transpose signs)
Alternative notation 2
(commas and semicolons, no transpose signs)

Operations

Matrix multiplication involves the action of multiplying each row vector of one matrix by each column vector of another matrix.

The dot product of two column vectors a and b is equivalent to the matrix product of the transpose of a with b,

By the symmetry of the dot product, the dot product of two column vectors a and b is also equivalent to the matrix product of the transpose of b with a,

The matrix product of a column and a row vector gives the outer product of two vectors a and b, an example of the more general tensor product. The matrix product of the column vector representation of a and the row vector representation of b gives the components of their dyadic product,

which is the transpose of the matrix product of the column vector representation of b and the row vector representation of a,

Matrix transformations

An n × n matrix M can represent a linear map and act on row and column vectors as the linear map's transformation matrix. For a row vector v, the product vM is another row vector p:

Another n × n matrix Q can act on p,

Then one can write t = p Q = v MQ, so the matrix product transformation MQ maps v directly to t. Continuing with row vectors, matrix transformations further reconfiguring n-space can be applied to the right of previous outputs.

When a column vector is transformed to another column vector under an n × n matrix action, the operation occurs to the left,

,

leading to the algebraic expression QM vT for the composed output from vT input. The matrix transformations mount up to the left in this use of a column vector for input to matrix transformation.

See also

Notes

  1. ^ Meyer (2000), p. 8

References

  • Axler, Sheldon Jay (1997), Linear Algebra Done Right (2nd ed.), Springer-Verlag, ISBN 0-387-98259-0
  • Lay, David C. (August 22, 2005), Linear Algebra and Its Applications (3rd ed.), Addison Wesley, ISBN 978-0-321-28713-7
  • Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), ISBN 978-0-89871-454-8, archived from the original on March 1, 2001
  • Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Brooks/Cole, ISBN 0-534-99845-3
  • Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
  • Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall