Understanding Matrix Multiplication and Function Composition
Explore the meaning and application of matrix multiplication in linear algebra and function composition in mathematics. Learn how to perform these operations, their relationship, and examples of reflection transformations.
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Presentation Transcript
Matrix Multiplication What does it mean?
Matrix Multiplication - Meaning Multiple Input ? = ?? b2 = c1 A A b1 b1 bp = c2 A b2 c2 cp = c1 ?? ?? = ? ?1 ?2 = cp A bp ??? = ??1 ??2
Matrix Multiplication - Meaning The notation g f is read as "g circle f ", "g round f ", "g about f ", "g composed with f ", "g after f ", "g following f ", "g of f", "f then g", or "g on f ". Composition Given two functions ? and g, the function ? ? . composition g f. is the ? = ? ? ? = ? ? ? g f g f ? = ? ? ? ? Matrix multiplication is the composition of two linear functions.
Matrix Multiplication - Meaning Composition ? ? ? ? ? ? A ? B ? = ?? ? = ?? ? C ? ? ?
Matrix Multiplication - Meaning 1 0 0 The first column of B ? ? ? ? ?1= ?1 A ??1 B ? = ?? ? = ?? 1 0 0 Input standard matrix The first column of C C ?1 ?1= ? ?
Matrix Multiplication - Meaning 0 1 0 The second column of B ? ? ? ? ?2= ?2 A ??2 B ? = ?? ? = ?? 0 1 0 Input standard matrix The second column of C C ?2 ?2= ? ?
? ? A ? B ? = ?? ? = ?? ? ? = ?? C ? The composition of A and B is ??? ? = ??1 ??2 = ?? Matrix Multiplication
Example reflection about the x-axis rotation by 180 R2 R2 R2 1 0 0 1 0 0 ? ? ? 1 1 ? = ?? ? = ?? ? ? reflection about the y-axis
Example 1 0 0 = 1 0 1 1 0 0 1 0 1 reflection about the x-axis rotation by 180 R2 R2 R2 1 0 0 1 0 0 ? ? ? 1 1 ? = ?? ? = ?? ?1 ?2 ?1 ?2 1 0 0 1 ? ? reflection about the y-axis
