Matrices. Solutions of Coupled Linear Ordinary Differential Equations. Functions of a Matrix. (10 Lectures) Cartesian Tensors Transformation of Co-ordinates. Einstein’s Summation Convention. Relation between Direction Cosines. Tensors. Algebra of Tensors. Sum, Difference and Product of Two Tensors. Contraction. Quotient Law of Tensors. Linear Algebra deals with linear equations and functions (i.e., have variables with maximum power of one. Example: ax+by+cz = d) and their representation using matrices and vectors. From matrices to vector spaces to linear transformations, you'll understand the key concepts and see how they relate to everything from genetics to nutrition to spotted owl extinction. Line up the basics? discover several different approaches to organizing numbers and equations, and solve systems of equations algebraically or with matrices. (b) (4 points) Let T: R3 → R3 denote the linear transformation that interchanges ~v 1 and ~v3 and has ~v2 as an eigenvector with eigenvalue −5. Write down [T]B, the matrix of T with respect to B. Answer: The matrix [T]B is gotten by writing down T(~v1), T(~v2), and T(~v3) in B coordinates and putting them as the columns of a matrix.

In this lecture, we discuss linear combinations and vector equations. Learn what matrices are and about their various uses: solving systems of equations, transforming shapes and vectors, and representing real-world situations. Learn how to add, subtract, and multiply matrices, and find the inverses of matrices. In this section we will give a brief review of matrices and vectors. We will look at arithmetic involving matrices and vectors, finding the inverse of a matrix, computing the determinant of a matrix, linearly dependent/independent vectors and converting systems of equations into matrix . Part 1. MATRICES AND LINEAR EQUATIONS 1 Chapter 1. SYSTEMS OF LINEAR EQUATIONS3 Background 3 Exercises 4 Problems 7 Answers to Odd-Numbered Exercises8 Chapter 2. ARITHMETIC OF MATRICES9 Background 9 Exercises 10 Problems 12 Answers to Odd-Numbered Exercises14 Chapter 3. ELEMENTARY MATRICES; DETERMINANTS

In this section we define some new operations involving vectors, and collect some basic properties of these operations. Begin by recalling our definition of a column vector as an ordered list of complex numbers, written vertically (Definition CV).The collection of all possible vectors of a fixed size is a commonly used set, so we start with its definition. So matrix addition takes two matrices of the same size and combines them (in a natural way!) to create a new matrix of the same size. Perhaps this is the “obvious” thing to do, but it does not relieve us from the obligation to state it carefully. Sensitivity of solution of linear equations. Let A be an invertible n x n matrix, and b and x be n-vectors satisfying Air = b. Suppose we perturb the jth entry of b by € +0 (which is a traditional symbol for a small quantity), so b becomes = b + cej. Matrices and Vectors. I’ve been asked by some curriculum writers to offer my thoughts on how I might introduced MATRICES and VECTORS to high-school students in response to the Common Core State Standards N-VM. In response I have done two things: i) Written a page chapter on introducing these concepts (TOC below.).