Why is row rank equal to column rank

Otherwise, it has dimension less than. As a consequence, the row rank of is less than or equal to its column rank. In a completely analogous manner, we prove that the column rank is less than or equal to the row rank: let Then, there exists a basis of row vectors that spans the same space spanned by the rows of. Denote by the matrix obtained from the vectors of the basis: Each row of can be expressed as a linear combination of. The coefficients of the linear combinations can be collected into a matrix such that The columns of the product are linear combinations of the columns of , with coefficients taken from.

So the span of the columns of is no larger than the span of the columns of because linear combinations of the columns of can be written as linear combinations of the columns of. There are columns in. As a consequence, the column rank of is less than or equal to its row rank. Thus, we have proved that and Therefore,.

Having proved that column and row rank coincide, we are now ready to provide the definition of rank. The rank of , denoted by , is defined as. In other words, the rank of a matrix is the dimension of the linear span of its columns, which coincides with the dimension of the linear span of its rows.

We have two possible cases. Active 1 month ago. Viewed 49k times. This looks like excessively complicated but I cannot think of any simpler explanation. Add a comment. Active Oldest Votes. This essential point of this argument is that elementary row operations, which by construction don't alter the row rank, also no not alter the column rank and similarly for column operations.

That this is so is because doing an elementary row operation just amounts to expressing all columns in a different basis. Also, isn't it true that you can have things other than zeros and ones in the reduced echelon form?

Therefore, everything off-diagonal would be zeroed out by a row operation or a column operation and what remain on the diagonal are ones and zeros. So I couldn't get the statement after equation in the answer.

That was what I thought when I wrote the answer. Show 4 more comments. Marc van Leeuwen Marc van Leeuwen k 6 6 gold badges silver badges bronze badges. Lonidard Lonidard 3, 1 1 gold badge 13 13 silver badges 25 25 bronze badges. You need of course a separate proof of the rank-nullity theorem, but this shows that the rank result is intimately related, in fact equivalent, to that theorem which student should know about anyway.

Of course the two options are equivalent, for reasons which I suppose boil down to OP's question. Now the four sentence proof. Dongryul Kim Dongryul Kim 6 6 silver badges 12 12 bronze badges. Rahul 1 1 silver badge 9 9 bronze badges.

Nick Alger Nick Alger Note: I think only left exactness of Hom is used here Phillip Williams Phillip Williams 2 2 bronze badges. James Short James Short 39 1 1 bronze badge. The best way is to take a concrete matrix. Take any matrix without loss of generality. That is Value 3 is a linear combination of 14 and 5. Value 10 is a linear combination of 13 and 4. This website is no longer maintained by Yu. ST is the new administrator. Linear Algebra Problems by Topics The list of linear algebra problems is available here.

Subscribe to Blog via Email Enter your email address to subscribe to this blog and receive notifications of new posts by email. Sponsored Links. Search for:. MathJax Mathematical equations are created by MathJax. Saurav answered Sep 3, Thanks for the link. But still I did not get why performing row operations won't change the column rank of the matrix.

At least It'll change the column space of the matrix. Next Qn. Related questions 2 votes. Why is the value of the determinant of adjoint of a matrix not equal to 1?