Study Sheet

Review the Homework. Quiz will have 3-4 problems, last 25 minutes.

Topics to know: all content from 2.4, 2.5, 5.1, and 5.2.

2.4: know what an invertible linear transformation is; know how to tell whether a linear transformation is an isomorphism; know how to prove that the inverse of a linear transformation is linear; know what an invertible matrix is; know that the matrix representation of a linear transformation is an invertible matrix iff the transformation is invertible; know what an isomorphism is; know how to prove that isomorphic vector spaces have the same dimension; know L(V, W) is isomorphic to M_mn(F) if dimV=n, and dimW=m; know why the giving of an ordered basis of V defines an isomorphism of F^n with V, and vice-versa; know the definition of row rank and column rank of a matrix (as per the handout) and that these are the same. Be able to prove that row rank of matrix is unchanged when a matrix is multiplied on either side (and hence both sides) by an invertible matrix (and the same for the row rank); you can find this material in the text in 3.2.

2.5: be able to change coordinates i.e. if T is a linear transformation, represented by matrix [T]_ab with respect to the bases a and b then know to compute [T]_gd from [T]_ab if the bases a and b are replaced by g and d. This is most important when a=b and g= d.

Know what it means for two matrices to be similar.

2.6: (For possible extra credit problems.) Know what the dual space is. Know what the dual basis is and be able to prove its existence. Know what the dimension of the dual space is.

5.1: Know what it means for a matrix to be diagonalizable; same for non-diagonalizable. Know how provide, with proof, some simple examples which are non-diagonalizable. Know how to find characteristic polynomial, eigenvalues, and eigenvectors of a matrix. Know that a matrix is diagonalizable iff there is a basis of eigenvectors (and proof if this statement). Know, given ordered basis of eigenvectors, how to construct a similarity to a diagonal matrix.

5.2: know how to prove that a set of eigenvectors belonging to distinct eigenvalues is a linearly independent set; know how to prove that if a matrix is diagonalizable then its characteristic polynomial splits over F (i.e. factors as a product of linear factors over F). Know that the converse of this statement is false and an example. Know what an eigenspace is. Know what the multiplicity of an eigenvalue is and be able to prove that the multiplicity of an eigenvalue is at most its multiplicity as a root of the characteristic polynomial, and that these two multiplicities are the same iff the linear transformation is diagonalizable.

5.2 (For possible extra credit problems): understand direct sums and be able to check whether a sum of several subspaces is direct. Know the fact that the sum of the different eigenspaces for T is always direct, and that T is diagonalizable iff V is the direct sum of the eigenspaces of T.

5.3 (for possible extra credit problems): know about limits of matrices; know criterion for sequence of powers of a fixed matrix to converge; know connection of convergence with eigenvalues and diagonalizability. Know what a transition ( also called stochastic) matrix is. Know that a power of a transition (or more generally the product two such) is a transition matrix, know what a probability vector is and that a product of a transition matrix and a probability vector is probability vector; know definition of a markov process (or chain).