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...Then the two compositions are ba algebra of linear transformations and ab matrices math djoyce fall products aren t same you can perform these on physical objects take we ve looked at operations addition a book first rotate it ip over start scalar multiplication again but rst ends used them to dene multipli up in dierent orientations cation for given basis v another w have an isomorphism matrix is associative al hom m vector spaces which n though s not commutative assigns transformation its that because corresponds composition standard functions any three wealso corre f g h ll show sponds if by showing sides values all x b dened so product be st while there few more things should look distributes with addi they re tion identity i multiplica since cancellation doesn work compute special kinds func powers square tions therefore transforma shouldn alternative proof would actually involve computations probably summation notation example let dimensional geometric something like rotation clo...