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When we perform row operations on the matrix, for instance, say we are reducing the matrix to a reduced row echelon form, how does doing this help us understand what the linearly independent columns are when we are clearly making changes to the rows?

  • You can think as a determinant. If the determinant is non-null (linearly independent), making linear changes will not make it null (linearly dependent). – Matheus Nunes Mar 19 '20 at 06:52
  • Think of the matrix as a system of linear equations. Why do we add equations on different lines to each other? – Brevan Ellefsen Mar 19 '20 at 06:53

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You have to know 2 things: First, doing a row operation on a matrix $A$ is the same as multiplying $A$ on the left by an invertible matrix $P.$ Second, if the columns of $A$ are $c_1,c_2, ... ,c_n,$ then the columns of $PA$ are $Pc_1,Pc_2, ... ,Pc_n$ so for scalars $r_1,r_2, ... ,r_n$ $$r_1c_1+r_2c_2+...+r_nc_n=0$$ iff $$r_1Pc_1+r_2Pc_2+...+r_nPc_n=0,$$ which is the same as saying that a relation of linear dependence is satisfied on the columns of $A$ iff the same relation is satisfied on the columns of $PA.$

P. Lawrence
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