A First Course in Linear Algebra · Problem RREF.M40

Systems of Linear Equations — Robert Beezer

Mathematics — Linear Algebra · reduced-row-echelon-form

Problem

Work carefully through Exercise RREF.M40, which is designed to illustrate the origins of the expression

[B]_{ij} = \sum_{k=1}^{m} \delta_{ik} [C]_{kj}

where

B

and

C

are row-equivalent matrices in reduced row-echelon form. Construct a small explicit example demonstrating that if

B

and

C

are row-equivalent

m \times n

matrices, then for each row

i

B

there exist scalars

\delta_{ik}

1 \le k \le m

, independent of

j

, such that this equation holds for all

1 \le j \le n