## MATH 221 May-June 2017 Lecture Summary

 We ek Date Contents 1 0515 First lecture Course outline and office hours, registration issues and timing §1.1 Systems of linear equations: definitions, Ex 1: a system of 2 equations of 2 unknowns; geometric interpretation, possibilities of the solution set. Matrix notation, Ex 2: redo Ex 1 in matrix form. Row operations. Ex 3: a system of 3 equations of 3 unknowns 0517 finished Ex 3, forward and backward phases, Ex 4 and Ex 5 §1.2 Row reduction and echelon forms: definitions of echelon form and reduced echelon form 0518 pivot position and pivot column, Ex 1,2; Finding general solution from reduced echelon form, Ex 2 again, Deciding existence and uniqueness from echelon form, Ex 3 §1.3 Vector equations: vectors, sum and scalar multiple, Ex 1, geometric meaning in R^2, linear combinations, Ex 2, 3 and 4, equivalence of being a linear combination and the linear system, Span, Ex 5. 0519 Ex 6 on Span §1.4 Matrix equations: product of a matrix and a vector, Ex 1 and 2. Equivalence of a matrix equation, a vector equation and a linear system, Ex 3. When does a matrix equation Ax=b has a solution for a given b? Ex 4. When does a matrix equation Ax=b has a solution for every b? Ex 5. Properties and computation of Ax, Ex 6 §1.5 Solution sets of Matrix equations: Ex 1 and 2, solution sets of homogeneous equations as span 2 0522 Victoria Day 0524 H1 due Ex 3, properties of the solution set of a homogeneous system. Nonhomogeneous system, Ex 4 and 5, properties of the solution set of a nonhomogeneous system, geometric meaning. 0525 §1.7 Linear independence: definition, Ex 1, 2, special cases, Ex 1 again, Ex 3, equivalence of linear dependence and existence of redundant vector, Ex 2 again, Ex 4, p>m implies linear dependence, Ex 5 §1.8 Linear transformations: terminologies, matrix transformations, Ex 1-4. 0526 Ex 5 on gemetric meaning of 2x2 matrix transformations. Linear transformations, definition and properties, Ex 6-9. §1.9 the matrix of a linear transformation: Theorem that any linear transformation from R^n to R^m is a matrix transformation, Ex 1-4, definition and theorem of onto and one-to-one linear transformations, Ex 5. 3 0529 §2.1 Matrix operations sum and scalar multiplication, Ex 1, composition of linear transformation and definition of matrix multiplication, Ex 2, computation rule, Ex 3, properties of matrix multiplication, Ex 4. Powers of square matrix, Ex 5, 6. Transpose, Ex 7, 8 §2.2 The inverse of a square matrix definition of invertible/inverse. Inverse is unique. Formula of the inverse matrix when n=2. 0531 Midterm exam 1 0601 Ex 1. Application for solving linear system, Ex 2. Properties of inverse matrix, Ex 3. Algorithm using row reduction, Ex 4 and 5. Right and left inverse for non-square matrices, Ex 6**. §2.3 Characterization of inverse matrices Invertible Matrix Theorem: on equivalent statements for invertible square matrices, Ex 1 and 2**, geometric meaning. Note: The exams will not include elementary matrices, right and left inverses. We also skip §2.4 0602 §2.5 Subspaces of R^n definition of subspaces, Ex 1, 2, 3. Column and null spaces of a matrix, Ex 4. Spanning set and basis. Ex 5--9. Pivot columns form a basis of the column space. §2.6 Dimension and rank Coordinate vector, Ex 1. Dimension theorem and definition, Ex 2. 4 0605 Ex 3. Rank. The rank theorem. The basis theorem. Extension of the Invertible Matrix Theorem. §3.1 Introduction to determinants Determiniant of a 3x3 matrix, Ex 1. Cofactor. Determiniant of an nxn matrix. The theorem on cofactor expansion. Ex 1 again, Ex 2. determinant of a triangular matrix, Ex 3. Check Online matrix calculator for Ex 2 §3.2 Properties of determinants theorem on change of the determinant under row/column operations, proof for 2x2 matrices 0607 H2 due Ex 1--4. A square matrix A is invertible iff det A \=0. Ex 5. det A^T = det A. det(AB) = det A . det B 0608 Ex 6. Interpretation of the determinant by considering it as a ratio of volumes (not in exam). §4.1 Eigenvectors and eigenvalues Ex 1, definition, Ex 2-4. Eigenvalues of triangular matrices, Ex 5. Eigenvectors with distinct eigenvalues are linearly independent, Ex 6. 0609 §4.2 The characteristic equation The characteristic equation, Ex 1-3, algebraic and geometric multiplicity, Ex 4-5, characteristic eq for triangular matrices. The following is covered in the final exam, but not in MT2 Discrete dynamical systems, Ex 6 (population dynamics), Ex 7 (Fibonacci's rabbits) 5 0612 Similarity, similar matrices have the same determinants, characteristic polynomials, eigenvalues, algebraic and geometric multiplicities of eigenvalues, Ex 8-9. A matrix with no real eigenvalue, Ex 10. §4.3 Diagonalization Ex 1,2. Diagonalizable matrix. An nxn matrix is diagonalizable if and only if it has n linearly independent eigenvectors. Ex 3-7. Special case when all eigenvalues are distinct. Ex 8. General case: An nxn matrix is diagonalizable if and only if the sum of geometric multiplicities is n. 0614 Midterm exam 2 0615 §4.4 Eigenvectors and linear transformations Matrix of a linear transformation in R^n relative to a basis, Ex 1-4. (We do not consider general linear transformations between vector spaces) §4.5 Complex eigenvalues Complex numbers, Ex 1-3, (link to Appendix B on complex numbers). Recall the matrix for a rotation. Complex eigenvalues, Ex 4 0616 Ex 4-7, conjugate eigenvalue, A=PCP^{-1} with C = [[a,-b], [b, a]] (We skip §4.6) §5.1 Inner product, length, and orthogonality dot product, Ex 1, length, Ex 2-3, distance, Ex 4, orthogonality, Ex 5, orthogonal to a set, orthogonal complement, Ex 6. 6 0619 H3 due Ex 6 again, Ex 7, angles, Ex 8 (We skip §5.1 Theorem 3) §5.2 Orthogonal sets orthogonal sets, Ex 1-2, an orthogonal set of nonzero vectors is linearly indepdent. Formula for coefficient. Ex 3. orthogonal projection onto a line, Ex 4-5. 0621 Ex 6. Orthonormal sets, Ex 7. Matrices with orthonormal columns, Ex 8. An mxn matrix U has orthonormal columns iff U^T U=I_n. In that case, U: R^n -> R^m preserves length and angle. Ex 9. Orthogonal/orthonormal matrices 0622 last lecture Ex 10 and remarks. §5.3 Orthogonal projections Theorem 8 on orthogonal decomposition, Ex 1. Theorem 9 on best approximation, Ex 2. Projection to subspaces with non-orthogonal basis, Ex 3. Review of December 2015 final exam. 0623 no class