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22A Practice Final

22A Practice Final - INFORMATION FOR FINAL Our nal will be...

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INFORMATION FOR FINAL Our final will be on Saturday, March 19th from 1:00-3:00pm, in Kleiber 3. The final is comprehensive . So you need to know everything covered this quarter. However, I will focus more on the part that has not been covered by the previous two midterms. Roughly, 45-50% of the final will be on material from chapters 20-25, and the rest will be on the material that has been tested before. Here is a list of the most important concepts from chapters 20-25. (Since I gave the lists of important concepts for the previous material already, I won’t repeat them here. So you should combine these lists together to study for the final.) (1) Change of basis and linear transformation using different bases. Understand definition of component vectors. Know how to find change of basis matrices and understand what they do. Understand the definition of associated matrices with respect to bases (other than the standard bases). Know how to find the associated matrix with respect to a new basis from the associated matrix with respect to an old basis. (2) Diagonalization. Know that an n × n matrix is diagonalizable if and only if it has n linearly independent eigenvectors, and know how to use this to determine whether a matrix is diagonalizable. Know how to express M as SDS - 1 if M is diagonalizable. Know the connection between diagonalizing M and describing linear transformations with different bases. (3) Orthogonal bases, orthonormal bases and Gram-Schmidt. Know the defi- nitions of orthogonal bases and orthonormal bases. Know how to use the Gram-Schmidt process to find an orthogonal/orthonormal basis for a given vector space. (4) Diagonalizing symmetric matrices and orthogonal matrices. Know that a real symmetric matrix is always diagonalizable. Know the results related to the eigenvalues and eigenvectors of real symmetric matrices. Know the definition of orthogonal matrices. Know how to express a real symmetric matrix as PDP - 1 where P is an orthogonal matrix.
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