LINEAR ALGEBRA REVIEW SHEET FOR THE FINAL EXAM The

LINEAR ALGEBRA: REVIEW SHEET FOR THE FINAL EXAM The problems on this exam will be chosen from the topics listed below. Calculus II is a prerequisite for this course and you should be prepared to deal with problems involving integration or differentiation of functions. If you think it may be helpful, feel free to bring an index card with integration formulas of your choice. (1) Basic matrix algebra. Multiplying matrices by scalars, matrix addition and multiplication, matrix transpose, matrix inversion, adjoint of a matrix and how it can be used to compute inverses of nonsingular matrices. Please make sure you are aware of the algebraic rules matrix operations obey (most are listed in section 1.3 of your text). These may be helpful in your review: 1.3 1, 2, 7, 20, 22 1.4 9, 10, 11, 12 2.3 1 (2) Symmetric, Hermitian, orthogonal and unitary matrices. Definitions, verifying that a matrix is symmetric, Hermitian, orthogonal or unitary. (3) Computing the determinant of a matrix; this includes both computing by definition and computing via Gaussian elimination. Related problems: 2.1 3, 4 2.2 1, 2, 12 (4) Properties of determinants (such as Theorem 2.2.2 and Theorem 2.2.3). Practice problems: 2.2 3, 6, 7 (5) Solving systems of equations using Cramer’s Rule. You may want to look at 2.3 2. (6) Vector spaces. The definition of vector spaces, common examples, verifying that a set with two operations is / is not a vector space. You may want to look at 3.1 3, 10, 11, 12, 13. (7) Subspaces. The definition of a subspace of a vector space, examples (such as subspaces of R2 and R3 ), verifying that a subset of a vector space is / is not a subspace. In particular, make sure you know how to prove that Span(v1 , v2 , ....., vn ), N (A), Ker(T ), Im(T ) and the eigenspaces corresponding to a given eigenvalue of T are all subspaces of the appropriate vector spaces. The following may be helpful: 3.2 2, 6, 8, 11, 18, 19, 20 2 (8) Linear combinations. Definition and examples of linear combinations, definition and geometric examples of a span of a set of vectors (see pages 128-129 of your textbook), definition of a spanning set. Some problems related to this: 3.2 9, 10, 14 (9) Linear independence. Definition and geometric interpretation, checking if a set of vectors is linearly (in)dependent. You may use these in your review: 3.3 1, 2, 4, 6, 15 (10) Basis and dimension. Definition and examples, verifying a set of vectors is / is not a basis. Please be able to do the verification both by definition and by using one of the following theorems: Theorem 3.4.1, Corollary 3.4.2, Theorem 3.4.3, Theorem 3.4.4. Some problems related to this: 3.4 3, 7, 11 and problems 1, 2, 7 from the Basis and Dimension extra homework sheet. (11) Finding coordinates of a vector with respect to a basis. This includes knowing how the coordinates change with a change of basis. Please note that in inner product spaces there is an efficient method for finding coordinates with respect to an orthonormal basis (see under orthonormal bases below). Problems for you to practice this: 3.5 4, 5, 6 (12) Nullspace, row space and column space of a matrix. Rank and nullity of a matrix. Finding basis and dimension of nullspaces, row spaces and column spaces, applying the column space technique to finding basis and dimension of a span of several vectors in Rn , relationship between rank and nullity of a matrix. You may want to look at 3.6 1, 2, 8, 11, 14, 15, 20. (13) Linear transformations (operators). Definition of linear transformations (operators), examples (see pages 176-179) of linear transformations (operators), verifying something is / is not a linear transformation (operator). The following may be of some help: 4.1 1, 4, 6, 8, 9, 11, 13, 16 (14) Kernel and image of a linear transformation. Definitions, finding basis and dimension of the kernel and the image of a given linear transformation, using the kernel to decide if a given linear transformation is one-to-one and / or onto. Here are some problems related to this: 4.1 17, 19, 23 3 (15) Matrix representations of linear transformations. Finding the matrix representation of a linear transformation T : V → W with respect to the bases EV (of V) and EW (of W), using the matrix representation to find Ker(T ), Im(T ) etc. You should also know how changing bases affects matrix representation. Practice problems: 4.2 2, 3, 5, 13, 14, 15, 18, 19 and 4.3 1, 4, 5. (16) Definition and examples of inner product spaces, verifying that something is an inner product (space). Please make sure you are able to verify that the inner products given on pages 245 and 246 of your textbook are indeed inner products. (17) Orthogonality. Orthogonality of vectors and / or subspaces, orthogonal complement of a subspace of an inner product space. Two very useful theorems here are Theorems 5.2.1 and 5.2.2 from your textbook. The following may be helpful: 5.2 2, 3, 4, 8 5.5 31 (18) Projections. Computing projections of a vector to another vector or to a subspace. This includes finding the best least squares approximation of a vector. Major results related to this are given in Theorems 5.5.7 and 5.5.8. You may need to use orthonormal sets in some of your computations. Problems for you to practice this: 5.1 3, 5, 9 5.4 2, 8 5.5 3, 27, 28, 30 (19) Orthonormal bases. Verifying something is an orthonormal basis, computing with orthonormal bases. Please make sure you know how to efficiently compute the coordinates of a vector with respect to an orthonormal basis, and how to efficiently compute inner products or magnitudes once the coordinates are known. You may use these in your review: 5.5 1, 2, 4, 5, 6, 7 (20) Gram-Schmidt process. Please note that Gram-Schmidt process is potentially necessary whenever one needs an orthonormal basis (and one often does in problems involving projections and diagonalization). Problems: 5.6 3, 4, 8 (21) Eigenvalues, eigenvectors and eigenspaces. Definitions, finding eigenvalues and eigenspaces (both for matrices and linear operators in general). Practice problems: homework sheet. 1, 2, 3, 4 from the Eigenvalues and Eigenvectors 4 (22) Diagonalization of diagonalizable matrices, diagonalization of symmetric (Hermitian) matrices using an orthogonal (unitary) diagonalizing matrix, applications (see class notes from Monday 12/5). Problems: 6.3 1, 2, 3 6.4 5 (23) Self-adjoint linear operators. Definition, verifying a linear operator is self-adjoint, the Spectral Theorem for self-adjoint linear operators. Related problems: 1, 2, 3 from the Self-adjoint Linear Operators homework sheet. (24) Complex (inner product) vector spaces. Basics of complex vector and inner product spaces. On this exam you may be asked to determine linear (in)dependence of vectors in a complex vector space, find basis and dimension of a complex vector space, study a linear transformation between two complex vector spaces or verify something is a complex inner product. For review please go over problems 8, 9, 10 on the Linear Algebra Over Complex Numbers homework sheet and problems 6.4 1, 2, 3, 9, 10 from your textbook.