L20100922

L20100922 - [THE CHANGE OF MATRIX & THE CHANGE OF...

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Unformatted text preview: [THE CHANGE OF MATRIX & THE CHANGE OF BASIS] September 22, 2010 The change of matrix Example The change of basis , a basis The coordinate of The coordinate of in in is is Let , is a basis of because Hence, is a basis of 1|Page [THE CHANGE OF MATRIX & THE CHANGE OF BASIS] September 22, 2010 is unchanged The coordinate of is the basis of The coordinate of in the basis of We also have We also have The key point is to understand the concept (for 2x2 matrix) 2|Page [THE CHANGE OF MATRIX & THE CHANGE OF BASIS] September 22, 2010 =================================Example================================== =========== is unchanged Let What is the corresponding matrix? Solution Now ’s coordinate is ’s coordinate is ’s coordinate is ’s coordinate is in in in in ’s coordinate is in 3|Page [THE CHANGE OF MATRIX & THE CHANGE OF BASIS] September 22, 2010 Then you have ===================================Example=========================================== is a basis of is a basis of The map we are going to write is What is the corresponding matrix? Solution From we get 4|Page [THE CHANGE OF MATRIX & THE CHANGE OF BASIS] September 22, 2010 From we get From we get So now what is the corresponding matrix? is a 2x3 matrix Why is it 2x3? Because only 2x3 matrix would give the answer in Solve matrix Solve for Solve for Solve for 5|Page [THE CHANGE OF MATRIX & THE CHANGE OF BASIS] September 22, 2010 Therefore Let's change the linear map and see what will happen... is still unchanged is another basis of There are 2 requirements for is to be basis of 1. The new basis has same number of element 2. You can express your new basis in term of your old basis If we can prove if 1. can be expressed in term of then the following is satisfied is linearly independent 2. It is maximized * * * * * * This tells you why is a basis 6|Page Why is this basis? It’s because [THE CHANGE OF MATRIX & THE CHANGE OF BASIS] September 22, 2010 Let's write linear map The corresponding matrix is So _________________________________________________________________________________ 1. Why do we have to change it to basis? Answer: Reason to change the basis is to make the A simple. 2. What will be the simplest A if there is no restriction for changing the basis? Answer: 3. What will be the simplest A if we can only choose the orthonormal basis with length 1? Term: orthonormal basis is 2 basis that are depending on each other, to be continue..... 7|Page ...
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This note was uploaded on 10/03/2010 for the course MATH 223 taught by Professor Loveys during the Spring '07 term at McGill.

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