MATH 220, Matrices Penn State University, University Park Spring 2022 CATALOG DESCRIPTION: MATH 220 (GQ) MATRICES (2 credits) Systems of linear equations appear everywhere in mathematics and its applications. MATH 220 will give students the basic tools necessary to analyze and understand such systems. The initial portion of the course teaches the fundamentals of solving linear systems. This requires the language and notation of matrices and fundamental techniques for working with matrices such as row and column operations, echelon form, and invertibility. The determinant of a matrix is also introduced; it gives a test for invertibility. In the second part of the course the key ideas of eigenvector and eigenvalue are developed. These allow one to ana lyze a complicated matrix problem into simpler components and appear in many disguises in physical problems. The course also introduces the concept of a vector space, a crucial element in future linear algebra courses. PREREQUISITE: MATH 110 or MATH 140 or MATH 140B or MATH 140E or MATH 140G or MATH 140H TEXTBOOK: MyLab Math with Pearson eText -- Instant Access -- for Linear Algebra and its Applications (18-Weeks) ISBN-13: 9780135851203 COURSE FORMAT: There are two 50-minute lectures each week. The sections covered in class are listed at the end of this syllabus. COURSE GOALS 1. Analyze and solve a system of linear equations. 2. Understand the important characteristics of matrices such as its fundamental subspaces, rank, determinant, eigenvalues and eigenvectors. 3. Learn how to use characteristics of a matrix to solve a system of linear equations or study properties of a linear transformation. 4. Understand important concepts of vector spaces such as independence, basis, dimension and orthogonality. 5. Use linear algebra to solve problems in branches of science, engineering and business. COURSE OBJECTIVES Section 1.1 – Systems of Linear Equations 1. Use elementary row operations to solve systems of linear equations. 2. Determine if a system of linear equations is consistent. 3. Determine the conditions for which a linear system is consistent 4. Determine the validity of statements about systems of linear equations, row operations, or matrices. Section 1.2 – Row Reduction and Echelon Forms 1. Identify matrices in echelon form and reduced echelon form. 2. Row reduce matrices to reduced echelon form. 3. Find the general solution to a system with a given augmented matrix. 4. Determine if a solution is consistent given a description of the corresponding coefficient matrix. 5. Determine the conditions for which a linear system has specified types of solutions. 6. Determine the validity of statements about row reduction and echelon forms. Section 1.3 – Vector Equations 1. Compute sums and scalar products of vectors, both algebraically and geometrically. 2. Convert between vector equations and systems of equations. 3. Determine if a vector is a linear combination of other vectors. 4. Characterize the span of a set of vectors algebraically or geometrically. 5. Determine the validity of statements about vectors and vector equations.Section 1.4 – The Matrix Equation Ax=b 1. Compute the product of a matrix and a vector. 2. Convert between matrix equations, vector equations, and systems of equations. 3. Solve matrix equations using augmented matrices. 4. Characterize the span of the column vectors of a matrix. 5. Determine whether a matrix equation has no solution, one solution, or infinitely many solutions. 6. Determine the validity of statements about vector equations and matrix equations. Section 1.5 – Solution Sets of Linear Systems 1. Determine if a system of equations has a nontrivial solution. 2. Solve a system of equations or a matrix equation and write the solution in parametric form. 3. Describe the solution sets of systems of equations geometrically. 4. Determine the validity of statements about solution sets of linear equations. Section 1.7 – Linear Independence 1. Determine if a set of vectors is linearly independent and determine if a vector is in a given span. 2. Determine conditions for which vectors are linearly independent or have a given span. 3. Determine the validity of statements about linear independence. Section 1.8 – Introduction to Linear Transformations 1. Algebraically find the image of a given vector under a linear transformation. 2. Given a linear transformation T(x)=Ax, find x for a given b in the image of T. 3. Determine the conditions for which a linear transformation has a given domain and codomain. 4. Determine if a vector is in the range of a linear transformation. 5. Geometrically describe the image of a vector under a linear transformation. 6. Use the linearity of transformations to find the images of vectors under the transformation. 7. Determine the validity of statements about linear transformations. Section 1.9 – The Matrix of a Linear Transformation 1. Find the standard matrix of a linear transformation. 2. Find vectors whose images under a linear transformation are given. 3. Determine the validity of statements about properties of linear transformations. 4. Determine if linear transformations are one-to-one or onto. Section 2.1 – Matrix Operations 1. Compute sums, products, and scalar products of matrices. 2. Find values of matrices such that products of matrices have given properties. 3. Determine the validity of statements about matrix operations. Section 2.2 – The Inverse of a Matrix 1. Find the inverse of a 2x2 matrix using the formula. 2. Use the inverse of a matrix to solve a linear system. 3. Determine the validity of statements about inverses of matrices. 4. Solve equations involving invertible matrices. 5. Find the inverse of a matrix using row reduction. Section 2.3 – Characterizations of Invertible Matrices 1. Determine if a given matrix is invertible. 2. Solve problems involving transformations and their matrices. Section 2.8 – Subspaces of Rn 1. Explain why a set is not a subspace of Rn. 2. Find a vector in a vector space or determine if a vector is in a vector space. 3. Find the dimension of the null and column spaces of a matrix. 4. Determine if a set of vectors is a basis for R2 or R3. 5. Determine the validity of statements about subspaces of Rn. 6. Find bases for the null and column spaces of a matrix.Section 2.9 – Dimension and Rank 1. Find a coordinate vector in a subspace 2.