※ 引述《iwantzzz (我愛大自然)》之銘言：
: suppose R and R' are 2x3 row-reduced echlon matrices and that the
: system RX=0 and R'X=0 have exactly the same solution. prove that
: R=R'.
By theorem: if A is an mxn matrix and m < n,
then AX = 0 has a non-trivial solution.
i.e. there is X≠0 s.t. RX = 0 and R'X =0.
Then we have three cases:
x
(i) X = ( y ) and x,y,z are in R (arbitrary)
z
0 0 0
=> if AX = 0 then A = ( ).
0 0 0
(ii) x = az and y,z are arbitrary (or y = bz and x,z are arbitrary),
1 0 -a 0 1 -b
=> if AX = 0 then A = ( ) (or A = ( )).
0 0 0 0 0 0
(iii) x = az and y = bz and z is arbitrary,
1 0 -a
=> if AX = 0 then A =( ).
0 1 -b
(note: a,b are constants and we assume A is a row-reduced echelon matrix.)
And RX = 0, R'X = 0 have exactly the same solutions,
hence no matter what the case is,
we have R = A = R'.
Q.E.D.
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