Added elementwise
Multiplied by a scalar
For matrices
the matrix product
coefficient matrix M of size m×n
“right-hand side” Y with n coordinates
Note the sizes: M:m×nX:n×1Y:m×1
M has size m×n
X is a column of length n — size n×1
MX is a column of length m — size m×1
fM sends a point X in n-dimensional space Rn to a point Y of m-dimensional space Rm. Notation: Rn→fMRm
The linear system MX=Y with M=1−115−433−32X=xyzY=uvw has solution xyz=−u+v−wu+v−2w3u−w or xyz=−11−111−230−1uvw=QuvwQ=−11−111−230−1
The execution of row operations shows that X=QYifMX=Y
Each elementary row operation can be reversed by another.
MX=Yif and only ifX=QY
Morever, MQY=MQY=MX=Yfor everyY=uvw
So fMQ=fM∘fQ=the identity mapandMQ=1=100010001
A slightly different matrix N=1−215−433−32
A square matrix
Its row echelon forms have two non-zero rows and one zero row
Row reduction for the linear system NX=Y=uvw(arbitrary right-hand side, as above) leads to (in the last row) 0=u+v−2w
Only one of the three given right-hand sides admits solutions
NX=111if and only ifX=−102+t−113anyt