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The '''[[row space]] and [[column space]]''' of an ''m''-by-''n'' [[matrix (mathematics)|matrix]] are the [[linear subspace]]s generated by row vectors and column vectors, respectively, of the matrix. Its [[Hamel dimension|dimension]] is equal to the [[Rank (linear algebra)|rank]] of the matrix and is at most min(''m'',&#8239;''n'').<ref>http://www.cliffsnotes.com/WileyCDA/CliffsReviewTopic/Row-Space-and-Column-Space-of-a-Matrix.topicArticleId-20807,articleId-20793.html</ref>
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The rest of article will consider matrices of [[real number]]s: row and column spaces are subspace of '''R'''<sup>''n''</sup> and '''R'''<sup>''m''</sup> [[real coordinate space|real spaces]] respectively. But row and column spaces can be constructed from matrices with components in any [[field (mathematics)|field]] and even a [[ring (mathematics)|ring]].
 
==Overview==
Let A be a ''m''-by-''n'' matrix. Then
# rank(A) = dim(rowsp(A)) = dim(colsp(A)),
# rank(A) = number of pivots in any echelon form of A,
# rank(A) = the maximum number of linearly independent rows or columns of A.
 
If one considers the matrix as a [[linear transformation]] from '''R'''<sup>''n''</sup> to '''R'''<sup>''m''</sup>, then the column space of the matrix equals the [[image (mathematics)|image]] of this linear transformation.
 
The column space of a matrix A is the set of all linear combinations of the columns in A. If A = ['''a'''<sub>1</sub>, ...., '''a'''<sub>n</sub>], then colsp(A) = span {'''a'''<sub>1</sub>, ...., '''a'''<sub>n</sub>}.
 
The concept of '''row space''' generalises to matrices to '''C''', the field of [[complex number]]s, or to any [[field (mathematics)|field]].
 
Intuitively, given a matrix '''A''', the action of the matrix '''A''' on a vector '''x''' will return a linear combination of the columns of '''A''' weighted by the coordinates of '''x''' as coefficients. Another way to look at this is that it will (1) first project '''x''' into the '''row space''' of A, (2) perform an invertible transformation, and (3) place the resulting vector '''y''' in the '''column space''' of '''A'''. Thus the result ''' y =A x''' must reside in the '''column space''' of A. See the [[singular value decomposition]] for more details on this second interpretation.
 
==Example==
Given a matrix J:
:<math>
  J =
  \begin{bmatrix}
    2 & 4 & 1 & 3 & 2\\
    -1 & -2 & 1 & 0 & 5\\
    1 & 6 & 2 & 2 & 2\\
    3 & 6 & 2 & 5 & 1
  \end{bmatrix}
</math>
the rows are
'''r'''<sub>1</sub> = (2,4,1,3,2),
'''r'''<sub>2</sub> = (−1,−2,1,0,5),
'''r'''<sub>3</sub> = (1,6,2,2,2),
'''r'''<sub>4</sub> = (3,6,2,5,1).
Consequently the row space of J is the subspace of '''R'''<sup>5</sup> [[linear span|spanned]] by { '''r'''<sub>1</sub>, '''r'''<sub>2</sub>, '''r'''<sub>3</sub>, '''r'''<sub>4</sub> }.  
Since these four row vectors are [[Linear independence|linearly independent]], the row space is 4-dimensional. Moreover in this case it can be seen that they are all [[orthogonality|orthogonal]] to the vector '''n''' = (6,−1,4,−4,0), so it can be deduced that the row space consists of all vectors in '''R'''<sup>5</sup> that are orthogonal to '''n'''.
 
==See also==
*[[Row space]]
*[[Column space]]
*[[null space]]
 
==External links==
{{wikibooks|Linear Algebra/Column and Row Spaces}}
*[http://mfile.akamai.com/7870/rm/mitstorage.download.akamai.com/7870/18/18.06/videolectures/strang-1806-lec06-20sep1999-80k.rm Lecture on column space and nullspace by Gilbert Strang of MIT]
*[http://wps.prenhall.com/am_leon_linearalg_7/53/13727/3514210.cw/content/index.html Row Space and Column Space]
 
==Notes==
{{reflist}}
 
[[Category:Abstract algebra]]
[[Category:Linear algebra]]

Revision as of 21:28, 3 March 2014

Lavender a aromatic herb grown for centuries and praised for fragrant, purplish blue floral arrangements. These small, drought-tolerant shrubs take full sun to partial shade outdoors. These people could take moderate water, might also survive in low water gardens once established (usually following a year). The flowers bloom on long, square-shaped stalks and buds can be up to two inches long.



In 97 we never did a trade unless there was confirming over all size. In 98 the volume was there anyway, the same as in 99. But, since then, just about every roll higher has been on what one could call "low volume" and frankly Certain pay that awful much attention to barefoot any other. The entire run up of 03 into 04 was accomplished on even less volume as compared to averages enjoy had us "need". Wait on the degree and miss the cruise. Nope, not gunna do it.

Clinton used the metaphor of the space between the invention of the club and the shield to describe the present situation involving war against terrorism. He was quoted saying "this gap needs to closed". Metaphors can give intangible concepts more impact with listeners.

Unless your carpet is incredibly worn or outdated, try steam cleaning it before replacing the item. If your goal end up being sell your house, to add to do is create a positive first impression. Often cleaning carpets will accomlish this goal, and replacing them becomes superfluous. Hiring a professional steam vapor cleaner can protect you from thousands over replacing carpet throughout an apartment.

One method for you to improve the exterior appearance of your home, in order to plant brightly colored flowers. Chose ones which might be appropriate per season. Further, trim the bushes and hedges, while keeping your lawn mowed and manicured. Easy improvements can instantly improve the entire look of the home.

The information mill not exactly as includes just five years ago families. This market is driven by program trades, not widespread participation as was always significantly in in the marketplace. In years gone buy, live people, making big decisions would make a move to get stocks, since they were buying other managers would see the experience and they'd buy so next on and so. Often volume and price appreciation "grew" on additional.

While dealing with these chores, try replacing your scenery. Adding birdhouses, sculptures and benches might give a little difference in your turf. Provided that you add feeders, some birds that'll be seen lurking into the garden are finches, woodpeckers and cardinals. With just a little planning, you can experience watching great view in the window. Even flagstones may entice that go to be able to your garden to analyse at the blooms of witch hazels or heath. You additionally think creating pathways made of old bricks, gravel or stone. Corporate ladder liven increase winter gardens without a sweat.

When you have any kind of issues with regards to exactly where in addition to how you can employ garden hedges, you can call us from our webpage.

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