Row-reduced matrices equivalences
Question
Solution
Short
Video
\(\LaTeX\)
No explanation / solution video to this exercise has yet been created.
Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Exercise:
Show that every mtimes n matrix A is row equivalent to a row-reduced matrix.
Solution:
Proof. Let Aa_ij leq ileq m leq j leq n a_ijin mathcalF. If all entries in the first row of A are then condition a first non-zero entry needs to be is satisfied for row #. If row # has a non-zero entry let k be the smallest index j for which a_j neq i.e. a_j forall leq j k and a_kneq . Multiply row # by fraca_k and condition a is satisfied for row #. Now for each leq i leq m add -a_ik times row # row i -a_ik R_+R_i longrightarrow R_i. We now turn to row #. If all entries in row # are we do nothing to this row. If some entry in row # is non-zero we multiply row # by a scalar element from mathcalF so that the leading term pivot in row # becomes . Recall that row # had a leading non-zero entry in col k. It follows that the leading non-zero term in row # can NOT be in col k. It must appear in a different col say in col k' where k'neq k. We add suitable multiples of row # to all the other rows till we get a matrix in which all entries in col k' are except the one in row #. In the last operations using row # we do NOT change any of the entries in row # that are in cols ...k. Also nothing will be changed in col k. Furthermore in case row # was the operations using row # will not affect row #. We now proceed to row # and continue doing the same procedure. After a finite number of elementary row operations we will arrive at a row-reduced matrix.
Show that every mtimes n matrix A is row equivalent to a row-reduced matrix.
Solution:
Proof. Let Aa_ij leq ileq m leq j leq n a_ijin mathcalF. If all entries in the first row of A are then condition a first non-zero entry needs to be is satisfied for row #. If row # has a non-zero entry let k be the smallest index j for which a_j neq i.e. a_j forall leq j k and a_kneq . Multiply row # by fraca_k and condition a is satisfied for row #. Now for each leq i leq m add -a_ik times row # row i -a_ik R_+R_i longrightarrow R_i. We now turn to row #. If all entries in row # are we do nothing to this row. If some entry in row # is non-zero we multiply row # by a scalar element from mathcalF so that the leading term pivot in row # becomes . Recall that row # had a leading non-zero entry in col k. It follows that the leading non-zero term in row # can NOT be in col k. It must appear in a different col say in col k' where k'neq k. We add suitable multiples of row # to all the other rows till we get a matrix in which all entries in col k' are except the one in row #. In the last operations using row # we do NOT change any of the entries in row # that are in cols ...k. Also nothing will be changed in col k. Furthermore in case row # was the operations using row # will not affect row #. We now proceed to row # and continue doing the same procedure. After a finite number of elementary row operations we will arrive at a row-reduced matrix.