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Descriptions of Transformations Represented by Matrices in Two Dimensions

In general a matrix operating on a vector may reflect, rotate, shear it, or a combination of these. We can deduce the nature of the transformation by inspection of the matrix.

  • If the determinant of the matrix is positive, it must be a rotation or a combination of a rotation and a shear.

  • It the determinant of the matrix is 1 and the columns or rows have length 1 when considered as a vector then it is a rotation. The matrix (in two dimensions) may be writtenWe can find the rotation angle by equating the transformation matrix to this and solving forNote that in general two one equation needs to be solved for each ofandto get the correct answer.

  • If the determinant of the matrix is negative, then it must be either a reflection or a combination of a reflection and a shear.

  • It the determinant of the matrix is -1 and the columns or rows have length 1 when considered as a vector then it is a reflection. The matrix (in two dimensions) may be writtenWe can find the anglethat the reflection line makes with the- axis by equating the transformation matrix to this and solving forNote that in general two one equation needs to be solved for each ofandto get the correct answer.

  • If the matrix scales a vector, it may scale in thedirection or thedirection or both. If it scales in thedirection only, then it must leave anyvalue unchanged, so that if the transformation is represented bythenso thatSinceandare arbitrary, we must haveandSimilarly, if the matrix represents a scaling in thedirection only, thenandOf course we may multiply matrices representingandscalings to obtain a matrix that scales in both directions simultaneously.

  • The area of any shape transformed is related to the area of the untransformed shape by the determinant of the transformation matrix,Ifandare the original and transformed shapes then