Year 12 Further Maths |
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Stretch Matrices How to recognise Stretch matrices have zeros in the two positions off the main diagonal Check that you agree the following, in which a general stretch matrix acts on the column vector (x, y) Notice that all x-values are now a times bigger, and that y-values are b times bigger. (Of course, you must substitute 'smaller' for 'bigger' if either of a or b are less than 1.) If a and b are equal, then the stretch is the same in all directions, and we have an enlargement of scale factor a. Application to combinations of transformations If you multiply a whole matrix by a stretching matrix, then the whole of the top row gets multiplied by the x-stretch and the whole of the bottom row by the y-stretch Notice that you can reverse engineer this, and do a kind of factorisation on the right-hand matrix in this equation: X is a factor of both elements in the top row, while Y is a factor of both elements in the bottom row. Under these circumstances you can 'factorise' as shown by the left-hand side of the equation. (Note that if there is no obvious common factor in one or other of the rows, then X or Y will have to be given the value 1 in the left-hand matrix.) |
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