linear algebra

Question

Rank of matrix :

1) Rank is equal to the number of linearly independent rows or columns in the matrix

2)Rank is equal to the number of non zero rows in ECHLON FORM of matrix

ARE BOTH the things same pls explain ?

Comments

• The dimension of row space and dimension of column space are same.
• The no of linearly independent columns is the dimension of column space as well as row space.
• After arriving at row reduced echelon form, the non-zero rows are the linearly independent rows and that number is equal to the dimension of row space. Which is also equal to the RANK

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This fact is trivial because as we move along finding out pivots and when we arrive at a new pivot position, both independent column nos. and independent row nos. gets incremented by $1$

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@ Debashish Deka 

pls take a matrix example for this .This will be helpful for me .

Answer 1

Rank is equal to the number of linearly independent rows or columns in the matrix 
 

The maximum number of linearly independent rows in a matrix A is called the row rank of A,

and the maximum number of linearly independent columns in A is called the column rank of A.

And  the column rank and the row rank are always equal.

Hence the rank of a matrix A is the dimension of the vector space generated  by its columns =  the dimension of the space generated by its rows .

Rank is equal to the number of non zero rows in ECHLON FORM of matrix OR BOTH the things same

Yes, both are same ...  the rank of a matrix in reduced row echelon form is equal to the number of non-zero rows it has .

That means row-equivalent matrices have the same row-rank.