Math 213 Exam 1 Study Guide PDF

MATH 213 EXAM 1 STUDY GUIDE General Information: The test will be in the Testing Center Wed. Oct. 6th till Fri. Oct. 8th with the 8th as a late day. The test will cover sections 1.1 through 3.3. A past exam and key can be found on Learning Suite. You may NOT use any notes, books, calculators, et...

MATH 213 EXAM 1 STUDY GUIDE General Information: The test will be in the Testing Center Wed. Oct. 6th till Fri. Oct. 8th with the 8th as a late day. The test will cover sections 1.1 through 3.3. A past exam and key can be found on Learning Suite. You may NOT use any notes, books, calculators, etc. Below is an outline of important points from each section. DISCLAIMER 1: This study guide is NOT intended to be an exhaustive list of everything that could possibly show up on the exam. It is a guide to help you focus your study on the most important things. DISCLAIMER 2: The practice exam is provided as a source of practice problems. You should not assume that your exam will have identical or extremely similar problems to what are on the practice exam. In general, simply working through the practice exam is NOT sufficient preparation for your exam. General Suggestions and Guidelines Go into the test calm and confident. You have worked hard and you know this material; this is just your chance to show off your knowledge. In general, simplify answers. However, I would suggest that once you get to a good answer, go on to the next problem, and then come back if you have time/energy and work on simplifying your answer. Do what you feel confident doing first. If you see a problem you don’t know how to approach, skip it, and come back later. On free response questions, you should show all your work. Show steps in a logical, easy to follow order. Ask yourself, could an average member of the class look at my work and follow what I am doing? (But don’t actually show it to an average member of the class while you are taking the test!) Use scratch paper. Some problems will not be obvious, and you may have to try a few approaches before you find the right one. Work through all this on scratch paper, then write up a clear solution on what you turn in. You might need a lot of scratch paper! Be sure to read the questions carefully and understand what they are asking. On all of your free response answers, ask yourself, have I answered the question that was asked? As much as you can, check your work, and think about whether your answer is reasonable. Especially with something like row reduction, it is very easy to make mistakes. 1.1 Be familiar with how to do addition and scalar multiplication of vectors, and the algebraic properties these satisfy. Also understand what these operations mean geometrically. Know the definition of linear combination and how to compute linear combinations of vectors. 1.2 Know the definition of dot product, how to compute dot products, and the algebraic properties. Know the definition of length and distance, and especially how these come from the dot product, and how to compute these. Be able to do simple proofs about norms and geometric properties related to do products by using algebraic properties of the dot product. Know how to compute angles using dot products, and the definition of orthogonal. Know the formula for the projection of a vector in the direction of another vector, and be able to use that formula, and understand what it means geometrically. 1.3 Be able to write an equation for a line or plane given a point and a normal vector, or the vector (parametric) form of the equation of a line given a point and a direction vector. 2.1 Understand what linear systems are and what solutions to systems mean visually. Know terminology like consistent and inconsistent. Know how to get the augmented matrix of a system of equations.  MATH 213 EXAM 1 STUDY GUIDE 2.2 Be able to recognize matrices in row echelon form and reduced row echelon form. Be able to identify leading entries or pivots. Know how to do row operations and when two matrices are row equivalent. Know the definition of rank and free variables and be able to recognize these once you have row reduced a matrix. Be really good at row reduction. Know how to translate a row-reduced matrix to the solution to its system, including writing the parametric vector form of solutions when there are infinitely many solutions, and recognizing when there are no solutions. Know the terminology homogeneous and trivial solution. 2.3 Understand how to think of a system of equations as a vector equation (solving for a vector as a linear combination of other vectors). Know the definition of span and be able to check if a vector is in the span of other vectors. Know how to check if a set of vectors spans all of Rn. Know the definitions of linearly independent and linearly dependent and dependence rela- tion (when they are linearly dependent). Be able to check if a set of vectors is linearly dependent or independent. Be familiar with the fact that a set is dependent if and only if one of the vectors can be written as a linear combination of the others. Know the fact that if there are more vectors than dimension, then the vectors are automatically linearly dependent. 3.1 Be familiar with addition and scalar multiplication of matrices. Know how the matrix product is defined and know how to multiply matrices. Know when a matrix product is defined, and the size of the product. Understand the matrix-column representation of matrix multiplication and understand that Ax is a linear combination of the columns of A. Understand how to take powers of a matrix. Know how to take the transpose of a matrix. Know the definition of a symmetric matrix. 3.2 Be familiar with the properties of matrix addition and scalar multiplication. Be familiar with the properties of matrix multiplication. Understand in particular that matrix multiplication is NOT commutative. Be especially careful with multiplying both sides of a matrix by something to cancel something that you get things on the correct side. Know what the identity matrix is and its special properties. Be familiar with properties of transpose. 3.3 Know and understand the definitions of invertible and inverse. Understand how to solve systems when the coefficient matrix is invertible. Understand the relation between invertibility and uniqueness of the solution to the corresponding system of equations. Know the formula for the inverse of a 2 × 2 matrix. Know the basic properties satisfied by inverses. Know the definition of an elementary matrix. Understand how elementary matrices relate to row operations, and how to find the inverse of an elementary matrix. Know the Fundamental Theorem of Invertible Matrices (understand this really well). Understand that you only need to check one side of the matrix multiplication to verify something is invertible (Theorem 3.13 in the book). Know how to find inverses using row reduction.