Math 2705.4 Eigenvectors and Linear TransformationsLet ! be an !-dimensional vector space, and let ! be an !-dimensional vector space, and let!: ! ! be a linear transformation where and ! denote bases for ! and ! respectively. ! with the coordinate vector of its image ! !Given any ! !, the coordinate vector !The connection between !is given by the following:and ! !!! ! :Let !! , !! , , !! be a basis for !.If ! = !! !! + !! !! + + !! !!!!!!Then ! = and ! ! = ! !! !! + !! !! + + !! !! = !! ! !! + !! ! !! + + !! ! !! because !!!is linear.Since the coordinate mapping from ! to ! is linear, we have:! !!= !! ! !!+ !! ! !!!!Since the !-coordinate vectors ! !! ! , ! !! ! ,, ! !!above can be written as a matrix equation, as follows:! !Where ! = ! !!!! !!!! !!!=! !!+ + !! ! !!!!are vectors in ! , the vector equation!!!!=! !! !!The matrix ! is called the _______________________________________.The matrix equation says that, so far as coordinate vectors are concerned, the action of ! on ! may beviewed as left-multiplic ...
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