COMPOSITION OF MAPPINGS

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## COMPOSITION OF MAPPINGS

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**COMPOSITION OF MAPPINGS**Matrix Multiplication Pamela Leutwyler**R3**R2 T S R2 ST**R3**Find the matrix for ST R2 T S R2 ST**One way to define the product of two**Matrices A and B is: THE nth COLUMN OF AB IS A( nth COLUMN OF B )**Because the matrix product AB represents a**COMPOSITION OF MAPPINGS, It is important that the RANGE OF B is within The DOMAIN OF A. If A is an mn matrix, It has m rows and n columns. Its domain is Rn, and its range is Rm If B is an np matrix, It has n rows and p columns. Its domain is Rp, and its range is Rn We can form the product AB And the answer will be An mp matrix. mn np**Conformable:**can be multiplied Consider the example: C D CD 32 24 34**Consider the example:**C D CD 32 24 34**Consider the example:**C D CD 32 24 34 To find the third column of CD take C times the third column of D**is row 2 of C DOT column 3 of D**Consider the example: C D CD 1 32 24 34 To break this down further, consider the second entry in column 3 the entry in row 2 column 3 of CD**In general, for any two conformable matrices A and B:**The entry in row j, column k of AB Is (row j of A) dot (column k of B) column k row j A B AB**Consider the example:**C D CD**Consider the example:**C D CD**Consider the example:**C D CD**Consider the example:**C D CD**Consider the example:**C D CD**Consider the example:**C D CD**Consider the example:**C D CD Sorry – I skipped a few steps!**Consider the example:**C D CD