MATH 262, Final exam review

Course topics

1. Linear equations
   1. Linear systems (1.1)
   2. Matrices, vectors and Gauss-Jordan elimination (1.2)
   3. The number of solutions of a linear system. Row-reduced echelon form (1.3)
   4. Matrix algebra (1.3)
2. Linear equations
   1. Linear transformatitions. Matrix of a linear transformation (2.1)
   2. Linear transformatitions in geometry (2.2)
   3. Invertible transformations. The inverse of a matrix (2.3)
   4. Matrix products (2.4)
3. Subspaces of Rⁿ and their dimension
   1. Image and kernel of a linear transformation. Linear span. (3.1)
   2. Linear relations, linear independence, redundant vectors (3.2)
   3. Basis of a subspace (3.2)
   4. The dimension of a subspace of Rⁿ (3.3)
   5. Coordinates. Matrices of linear transformations (3.4)
   6. Similar matrices. Change of basis in Rⁿ (3.4)
4. Vector spaces
   1. Vector spaces. Subspaces. Examples (4.1)
   2. Linear transformations. Isomorphisms (4.2)
   3. Coordinates. Matrices of linear transformations (4.3; skip "Change of basis for general vector spaces")
5. Orthogonality
   1. Dot product. Orthogonal vectors, orthogonal projections onto subspaces (5.1)
   2. Orthonormal bases, orthogonal complement (5.1)
   3. Pythagorean theorem and Cauchy's inequality (5.1)
   4. Gram-Schmidt process and QR-factorization (5.2)
   5. Orthogonal matrices, orthogonal transfromations (5.3)
6. Determinants and introduction to eigenvalue problems
   1. Determinants; geometrical interpretation in the 2x2 and 3x3 cases (6.1)
   2. Laplace's rule (cofactor expansion) (6.1)
   3. Determinants and the Gauss-Jordan elimination (6.2)
   4. Properties of determinants: products, similar matrices, inverse, transpose (6.2)
   5. Eigenvalues and eigenvectors (7.1, 7.2)

Objectives

In addition to Test 1 and Test 2:

• Compute QR factorization of a matrix (perform Gram-Schmidt process)
• Find eigenvalues and eigenvectors of a matrix
• Compute determinants in (at least) three ways: definition, Laplace's rule, Gauss elimination and be able to use their properties
• Use basic concepts of geometry in Rⁿ: dot product, orthogonal vectors, orthogonal projections, Pythagorean theorem, Cauchy inequality, orthogonal complements
• Use properties of orthogonal matrices and orthogonal transformations

Technical stuff/formulas to memorize

In addition to Test 1 and Test 2:

• The Gram-Schmidt algorithm and the QR factorization
• Determinants for 2x2 and 3x3 matrices ("Sarrus rule" or "triangle rule")
• Laplace's rule (cofactor expansion)
• Characteristic equation det(A-λI)=0.

Review questions

(At the end of each chapter)

Chapter 5: 1, 2, 4, 6, 7, 9, 10, 12, 13, 15-17, 25, 29.

Chapter 6: 1-7, 9-16, 18-21, 28.

Chapter 7: 1-3, 5, 6, 8-10, 13, 30, 33, 46, 47.

Format of the exam

The format of the exam will be similar to that of the midterms. The duration of the exam is 2 hours, and the length of the exam will be approximately twice that of the midterms. The final exam is closed books/notes, and the usual rules about calculator use apply (you should not need calculators to solve problems on the exam anyway). A number of questions on the test will be simple yes/no type questions, which may be taken from the review exercises in the end of each chapter.