corresponding to the blue parallelogram the linear operation that transforms the red to the blue parallelogram can be thought of as succession of two linear transformations the first is to transform the coordinates of points on the red parallelogram (in standard basisto the basis formed by the columns of secondrelative to this new basiswe apply the matrix above note that the input of this matrix is with respect to the new basiswhereas the output is with respect to the standard basis the matrix for the combined operation is now # - - - ba - / - / - which maps [ ]to [ ](does not changeand [ ]to [ - ]we say that we apply shear in the direction [ ]the significance of such an operation is that shear does not alter the volume of the parallelogram the second (blueparallelogram has an easier formbecause one of the sides is parallel to the -axis by applying another shearin the direction [ - ]we can obtain simple (greenrectanglewhose volume is in matrix termswe add / times the second column of to the first column of bto obtain the matrix - which is diagonal matrixwhose determinant is - corresponding to the volume of all the parallelograms theorem summarizes number of useful matrix rules for the concepts that we have discussed so far we leave the proofswhich typically involves "writing outthe equationsas an exercise for the readersee also [ theorem useful matrix rules (ab)ba (ab)- - - ( - )( )- = - det(abdet(adet( xax tr axxq det(ai aii if [ai is triangular nextconsider an matrix for which the matrix inverse fails to exist that isa is either non-square ( por its determinant is instead of the inversewe can use its so-called pseudo-inversewhich always exists shear