The following is a chapter by chapter guide intended to help you organize the material we have covered in class as you study for your exam. This exam will not explicitly test material from sections covered before the first midterm, but it is still important to know the earlier material as later chapters build on this material. The information here is only intended to serve as a guidline, and may not explicitly mention everything that you need to study. Please review all homework, quiz and example problems for the chapters given below (including the true/false problems). I have also compiled a list of additional practice problems.
2.7: You should know what it means for a transformation to be linear, and how to find the standard matrix of a linear transformation.
2.8: You should also be able to determine whether or not a linear transformation is one-to-one, onto and/or invertible. The table on p. 188 provides a good summary of the important ideas from this section.
4.1: You should know how to determine whether a given set is a subspace or not. Remember that all three conditions must hold for a set to be a subspace, but a single counterexample will show that it is not. Make sure you understand all the special subspaces associated to a matrix. You may use the fact that any set that can be written as a special subspace of a matrix is a subspace.
4.2: You need to understand what a basis is and what the dimension of a subspace is. Pay special attention to the steps for showing that a given set is a basis for a subspace of 𝓡n.
4.3: You should know how to find a basis and the dimensions of the subspaces associated to a matrix. This is summarized in the table and the blue box on p. 259, and it is important that you know all of the information contained in them.
3.1: You should know how to find the determinant of a matrix using a cofactor expansion.
3.2: You need to know all the properties of determinants listed in this section, and you should be able to find the determinant of a matrix using row operations. You can ignore the section on Cramer’s Rule.
5.1: This is an extremely important chapter! You must know how to find the eigenvalues and eigenvectors of a matrix, and a basis for the eigenspace corresponding to an eigenvalue.
5.2: You should know how to find the characteristic equation of a matrix, both directly from the matrix and given information about eigenvalues and eigenvectors. You should know how to use the characteristic equation to determine eigenvalues and multiplicities.
5.3: This is another very important chapter. You need to know how to determine whether a matrix is diagonalizable and how to find the diagonalization if it is. You should be able to use this to compute high powers of a diagonalizable matrix.
Maintained by ynaqvi and last modified 04/02/13