Review of linear algebra

Save this PDF as:
Size: px
Start display at page:

Download "Review of linear algebra"

Transcription

1 Review of linear algebra 1 Vectors and matrices We will just touch very briefly on certain aspects of linear algebra, most of which should be familiar. Recall that we deal with vectors, i.e. elements of R n, which here we will denote with bold face letters such as v, and scalars, in other words elements of R. We could also use C n, or Q n as needed, with scalars respectively C or Q. The main point is that, for the scalars, we need to be able to add, subtract, multiply and divide (except by 0). We can then add two vectors: if v = (v 1,..., v n ) and w = (w 1,..., w n ), then v + w = (v 1 + w 1,..., v n + w n ). Scalar multiplication is similarly defined: given t R and v = (v 1,..., v n ) R n, tv = (tv 1,..., tv n ). Vector addition is commutative and associative, there is a zero vector 0 = (0,..., 0), and every vector v has an additive inverse v = ( 1)v = ( v 1,..., v n ). Scalar multiplication satisfies: for all s, t R and v R n, s(tv) = (st)v and 1v = v. Finally there are two analogues of the distributive law: s, t R and v R n, (s + t)v = sv + tv, and for all t R and v, w R n, t(v + w) = tv + tw. These are easily check by using the usual properties of addition and multiplication for real numbers. We will not discuss the standard definition of linear independence, span, dimension or basis here. However, we will frequently use the standard basis e 1,..., e n, where the components of e i are 0 except for the i th component, which is 1. Thus, any vector v = (v 1,..., v n ) can be uniquely written in terms of the standard basis: v = n i=1 v ie i. There is also the dot product or scalar product or inner product of two vectors v, w R n, which we shall call the inner product and write as v, w, although if called the dot product it is usually written as v w. Note that the product of two vectors is a scalar, whence the name scalar product. It 1

2 is bilinear and symmetric: for all v, w, u R n and t R, v + w, u = v, u + w, u ; tv, w = t v, w ; u, v + w = u, v + u, w ; v, tw = t v, w ; v, w = w, v Of course, the third and fourth identities are a consequence of the first two and the symmetry condition. We can also define the length or norm of v: v = ( v, v ) 1/2. The standard basis e 1,..., e n is orthonormal: for all i, j, { 0, if i j; e i, e j = 1, if i = j. Any basis u 1,..., u n with this property, that u i, u j = 1 if i j and u i, u i = u i 2 = 1, will be called an orthonormal basis. Our main interest will be interesting sets of matrices. Recall that an m n matrix is a rectangular array A = a 11 a a 1n a m1 a m2... a mn We will often abbreviate this as A = (a ij ). The above matrix consists of m rows and n columns. We refer to the number a ij as the ij th entry. This means that a ij is the number in the i th row and j th column. In particular a vector (x 1,..., x n ) is also a matrix, in this case a 1 n matrix. We will call such a matrix a row vector. We can also think of a vector as an n 1 matrix, which we shall refer to as a column vector. (We will often have to think of vectors as column vectors because of our conventions on the way we write functions.) The set of all m n matrices is written M m,n (R). M m,n (C), M m,n (Q), and even M m,n (Z) are defined similarly. We can add two matrices in M m,n (R) and multiply a matrix by a scalar. The zero matrix O = O m,n M m,n (R) is the matrix all of whose entries are 0. In case m = n, we abbreviate M n,n (R) by M n (R) and call such a matrix a square (n n) matrix. An important element of M n (R) is the identity matrix I n = I, 2.

3 whose diagonal entries a ii are equal to 1 and whose other entries a ij, i j, are equal to 0. It is easy to see that, for all A M m,n (R), I m A = AI n = A. Given an m n matrix A and an n k matrix B, we can form the matrix product AB, an m k matrix whose ij th entry is given by n t=1 a itb tj. Thus the ij th entry is the inner product of the i th row of A with the j th column of B. Matrix multiplication is associative and distributes over matrix addition, where defined, but for A, B M n (R) (the only case where AB and BA are both defined and of the same shape), it is rarely the case that AB = BA: matrix multiplication is not commutative. Recall that a linear function F : R n R m is a function F such that, for all v, w R n and t R, F (v + w) = F (v) + F (w) and F (tv) = tf (v). A linear function is completely specified by its values on the standard basis vectors e 1,..., e n. Conversely, given any set of vectors v 1,..., v n R m, there is a unique linear function F : R n R m such that F (e i ) = v i for all i, namely F (x 1,..., x n ) = i x iv i. In this case, recall that we can associate an m n matrix to F as follows: write the vectors v i = (a 1i,..., a mi ). Then to F we associate the matrix A = a 11 a a 1n a m1 a m2... a mn. Here the columns of A are the vectors v i, written vertically, and the linear map F (x 1,..., x n ) corresponds to the matrix product A x, where A x is the n 1 matrix (column vector) whose j th entry is n i=1 a jix i. In particular A e i = v i, written as a column vector; its j th entry is a ji and it is equal to k j=1 a jie j, where in the equality A e i = k a ji e j the e i on the left is a basis vector in R n and the e j on the right is a basis vector in R k. Note the reversal of the indices! The case F : R n R n corresponds to square (n n) matrices. For example the linear function Id R n corresponds to the identity matrix I n. Then we have: Proposition 1.1. If G: R k R n and F : R n R m are linear maps, and A and B are the matrices corresponding to F and G respectively, then F G is again linear and the matrix corresponding to F G is the matrix product A B. j=1 3

4 This gives another, more conceptual proof of the associativity of matrix multiplication. Let A be an m n matrix A = (a ij ). Recall that the transpose matrix t A is the n m matrix whose (i, j) th entry is a ji. For example, if A is a square (n n) matrix, then t A is the reflection of A along the diagonal running from upper left to lower right. Clearly, t ( t A) = A. A calculation shows that, for all standard basis vectors e i R m and e j R n, e i, Ae j = t Ae i, e j. (Here of course the first inner product is of vectors in R m and the second is of vectors in R n.) Using bilinearity, it follows that, for all v R m and w R n, v, Aw = t Av, w. From this (or directly from the definitions) one can prove: If A is an m n matrix and B is an n k matrix, then t (AB) = t B t A. 2 Invertible matrices We will write linear maps F : R n R m as matrices A: F (v) = Av, with the understanding that, for the right hand side, v must be viewed as a column vector. Define the nullspace or kernel of A to be the set {v R n : Av = 0}. We then have the basic result: Proposition 2.1. The linear function A: R n R m is injective the nullspace of A is {0} the columns of A are linearly independent. The linear function A: R n R m is surjective the columns of A span R m. Corollary 2.2. Let F : R n R m be a linear function, corresponding to the matrix A. (i) If F is injective, then n m. (ii) If F is surjective, then n m. (iii) If n = m, then F is injective F is surjective F is a bijection, and in this case the inverse function F 1 corresponds to a matrix, denoted A 1, with the property that AA 1 = A 1 A = I n. 4

5 We call a matrix A M n (R) invertible if an inverse A 1 exists. The problem of deciding when a given n n matrix A is invertible can be answered by determinants. Recall that, for every n, we have a function det: M n (R) R with the following properties: 1. For all A, B M n (R), det(ab) = (det A)(det B). 2. det I n = A is invertible det A 0. In this case, 4. det t A = det A. det(a 1 ) = (det A) 1. Define the general linear group GL n (R) to be the subset of M n (R) consisting of invertible matrices. Equivalently, by (3) above, GL n (R) = {A M n (R) : det A 0}. The subset GL n (R) of M n (R) is closed under products, I n GL n (R), and if A GL n (R), then by definition A 1 exists and A 1 GL n (R); note that A 1 is invertible and that (A 1 ) 1 = A. Define the special linear group SL n (R) via: SL n (R) = {A M n (R) : det A = 1}. Clearly SL n (R) GL n (R). By (1) above, SL n (R) is closed under multiplication and by (2) above, I n SL n (R). Finally, if A SL n (R), then A is invertible and A 1 SL n (R) by (3). 3 Orthogonal matrices Orthogonal matrices are invertible matrices with very special geometric properties. Definition 3.1. A linear function A: R n R n is an isometry if, for all v R n, Av = v. In other words, A preserves length. Proposition 3.2. Given A M n (R), the following conditions on A are equivalent. (i) A is an isometry, i.e. for all v R n, Av = v. 5

6 (ii) For all v, w R n, Av, Aw = v, w. In other words, A preserves inner product. (iii) The columns of A are an orthonormal basis of R n. (iv) A is invertible and t A = A 1. (v) The rows of A are an orthonormal basis of R n. Proof. (i) = (ii): This follows from the polarization identity: For all v, w R n, v + w 2 v w 2 = 4 v, w. This in turn follows from the bilinearity and symmetry of inner product and expansion: For example, v + w 2 = v + w, v + w = v, v + 2 v, w + w, w, and similarly for v w 2. Then, if A is an isometry, 4 Av, Aw = Av + Aw 2 Av Aw 2 Hence Av, Aw = v, w. = A(v + w) 2 A(v w) 2 = v + w 2 v w 2 = 4 v, w. (ii) = (i): If Av, Aw = v, w for all v, w R n, then take v = w, so that Av 2 = Av, Av = v, v = v 2. (ii) = (iii): The columns of A are equal to u i = Ae i. By (ii), u i, u j = Ae i, Ae j = e i, e j. Thus u 1,..., u n is an orthonormal basis of R n. (iii) (iv): The ij th entry of t AA is the inner product u i, u j. Hence t AA = I n u i, u j is 0 if i j and 1 if i = j the columns of A are an orthonormal basis of R n. (iv) (v): Similar to the above, using A t A instead of t AA. (iv) = (ii): If t A = A 1, then for all v, w R n, Av, Aw = v, t AAw = v, A 1 Aw = v, w. We see that any of the five statements in the proposition implies any other, so they are all equivalent. 6

7 Definition 3.3. A matrix A M n (R) satisfying any (and hence all) of the equivalent properties above is called an orthogonal matrix. The set of all orthogonal n n matrices is denoted O n, the orthogonal group. The set of all orthogonal matrices with determinant 1 is denoted SO n, the special orthogonal group. Proposition 3.4. If A, B O n, then AB O n. Moreover I n SO n and hence I n O n. Finally, if A O n, then A 1 O n. Similar statements hold with O n replaced by SO n. Proof. If A, B O n, then t (AB) = t B t A = B 1 A 1 = (AB) 1. Thus AB O n. Clearly I n SO n. Finally, note that, in general, if A is an n n matrix with an inverse A 1, then t (A 1 ) = ( t A) 1, by applying the identity t (AB) = t B t A to the product AA 1 = I. Thus, if A is orthogonal, It follows that A 1 O n. t (A 1 ) = ( t A) 1 = (A 1 ) 1. The following says that there is not a big difference between O n and SO n : Proposition 3.5. If A O n, then det A = ±1. Proof. Using t A = A 1, we see that det A = det t A = det A 1 = (det A) 1. Thus (det A) 2 = 1, so that det A = ±1. We sometimes think of SO n as the set of rigid motions of R n (fixing the origin). More details about SO 2 and O 2 are in the homework. 7

Linear Algebra V = T = ( 4 3 ).

Linear Algebra V = T = ( 4 3 ). Linear Algebra Vectors A column vector is a list of numbers stored vertically The dimension of a column vector is the number of values in the vector W is a -dimensional column vector and V is a 5-dimensional

More information

Review of Linear Algebra

Review of Linear Algebra Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=

More information

Elementary maths for GMT

Elementary maths for GMT Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1

More information

Linear Algebra Review. Vectors

Linear Algebra Review. Vectors Linear Algebra Review 9/4/7 Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa (UCSD) Cogsci 8F Linear Algebra review Vectors

More information

Math Linear Algebra Final Exam Review Sheet

Math Linear Algebra Final Exam Review Sheet Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of

More information

Knowledge Discovery and Data Mining 1 (VO) ( )

Knowledge Discovery and Data Mining 1 (VO) ( ) Knowledge Discovery and Data Mining 1 (VO) (707.003) Review of Linear Algebra Denis Helic KTI, TU Graz Oct 9, 2014 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, 2014 1 / 74 Big picture: KDDM Probability Theory

More information

Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the

More information

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

More information

Lecture 3: Matrix and Matrix Operations

Lecture 3: Matrix and Matrix Operations Lecture 3: Matrix and Matrix Operations Representation, row vector, column vector, element of a matrix. Examples of matrix representations Tables and spreadsheets Scalar-Matrix operation: Scaling a matrix

More information

[ Here 21 is the dot product of (3, 1, 2, 5) with (2, 3, 1, 2), and 31 is the dot product of

[ Here 21 is the dot product of (3, 1, 2, 5) with (2, 3, 1, 2), and 31 is the dot product of . Matrices A matrix is any rectangular array of numbers. For example 3 5 6 4 8 3 3 is 3 4 matrix, i.e. a rectangular array of numbers with three rows four columns. We usually use capital letters for matrices,

More information

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

More information

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.

More information

2. Every linear system with the same number of equations as unknowns has a unique solution.

2. Every linear system with the same number of equations as unknowns has a unique solution. 1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations

More information

A = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].

A = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3]. Appendix : A Very Brief Linear ALgebra Review Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics Very often in this course we study the shapes

More information

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and

A matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.

More information

LS.1 Review of Linear Algebra

LS.1 Review of Linear Algebra LS. LINEAR SYSTEMS LS.1 Review of Linear Algebra In these notes, we will investigate a way of handling a linear system of ODE s directly, instead of using elimination to reduce it to a single higher-order

More information

Conceptual Questions for Review

Conceptual Questions for Review Conceptual Questions for Review Chapter 1 1.1 Which vectors are linear combinations of v = (3, 1) and w = (4, 3)? 1.2 Compare the dot product of v = (3, 1) and w = (4, 3) to the product of their lengths.

More information

Quantum Computing Lecture 2. Review of Linear Algebra

Quantum Computing Lecture 2. Review of Linear Algebra Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces

More information

1. Foundations of Numerics from Advanced Mathematics. Linear Algebra

1. Foundations of Numerics from Advanced Mathematics. Linear Algebra Foundations of Numerics from Advanced Mathematics Linear Algebra Linear Algebra, October 23, 22 Linear Algebra Mathematical Structures a mathematical structure consists of one or several sets and one or

More information

CS 246 Review of Linear Algebra 01/17/19

CS 246 Review of Linear Algebra 01/17/19 1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector

More information

Matrix Algebra Determinant, Inverse matrix. Matrices. A. Fabretti. Mathematics 2 A.Y. 2015/2016. A. Fabretti Matrices

Matrix Algebra Determinant, Inverse matrix. Matrices. A. Fabretti. Mathematics 2 A.Y. 2015/2016. A. Fabretti Matrices Matrices A. Fabretti Mathematics 2 A.Y. 2015/2016 Table of contents Matrix Algebra Determinant Inverse Matrix Introduction A matrix is a rectangular array of numbers. The size of a matrix is indicated

More information

MATH 2030: MATRICES ,, a m1 a m2 a mn If the columns of A are the vectors a 1, a 2,...,a n ; A is represented as A 1. .

MATH 2030: MATRICES ,, a m1 a m2 a mn If the columns of A are the vectors a 1, a 2,...,a n ; A is represented as A 1. . MATH 030: MATRICES Matrix Operations We have seen how matrices and the operations on them originated from our study of linear equations In this chapter we study matrices explicitely Definition 01 A matrix

More information

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing

More information

Matrix Algebra: Summary

Matrix Algebra: Summary May, 27 Appendix E Matrix Algebra: Summary ontents E. Vectors and Matrtices.......................... 2 E.. Notation.................................. 2 E..2 Special Types of Vectors.........................

More information

Chapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in

Chapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in Chapter 4 - MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a 12... a 1j... a 1n a 21. a 22.... a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column

More information

Math 320, spring 2011 before the first midterm

Math 320, spring 2011 before the first midterm Math 320, spring 2011 before the first midterm Typical Exam Problems 1 Consider the linear system of equations 2x 1 + 3x 2 2x 3 + x 4 = y 1 x 1 + 3x 2 2x 3 + 2x 4 = y 2 x 1 + 2x 3 x 4 = y 3 where x 1,,

More information

Matrices and Linear Algebra

Matrices and Linear Algebra Contents Quantitative methods for Economics and Business University of Ferrara Academic year 2017-2018 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2

More information

A PRIMER ON SESQUILINEAR FORMS

A PRIMER ON SESQUILINEAR FORMS A PRIMER ON SESQUILINEAR FORMS BRIAN OSSERMAN This is an alternative presentation of most of the material from 8., 8.2, 8.3, 8.4, 8.5 and 8.8 of Artin s book. Any terminology (such as sesquilinear form

More information

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing

More information

Chapter 2 Notes, Linear Algebra 5e Lay

Chapter 2 Notes, Linear Algebra 5e Lay Contents.1 Operations with Matrices..................................1.1 Addition and Subtraction.............................1. Multiplication by a scalar............................ 3.1.3 Multiplication

More information

A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 2010

A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 2010 A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 00 Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics

More information

Basic Concepts in Linear Algebra

Basic Concepts in Linear Algebra Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University February 2, 2015 Grady B Wright Linear Algebra Basics February 2, 2015 1 / 39 Numerical Linear Algebra Linear

More information

MATH Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product.

MATH Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product. MATH 311-504 Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product. Determinant is a scalar assigned to each square matrix. Notation. The determinant of a matrix A = (a ij

More information

Determinants Chapter 3 of Lay

Determinants Chapter 3 of Lay Determinants Chapter of Lay Dr. Doreen De Leon Math 152, Fall 201 1 Introduction to Determinants Section.1 of Lay Given a square matrix A = [a ij, the determinant of A is denoted by det A or a 11 a 1j

More information

Math 443 Differential Geometry Spring Handout 3: Bilinear and Quadratic Forms This handout should be read just before Chapter 4 of the textbook.

Math 443 Differential Geometry Spring Handout 3: Bilinear and Quadratic Forms This handout should be read just before Chapter 4 of the textbook. Math 443 Differential Geometry Spring 2013 Handout 3: Bilinear and Quadratic Forms This handout should be read just before Chapter 4 of the textbook. Endomorphisms of a Vector Space This handout discusses

More information

NOTES on LINEAR ALGEBRA 1

NOTES on LINEAR ALGEBRA 1 School of Economics, Management and Statistics University of Bologna Academic Year 207/8 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura

More information

Matrix Arithmetic. j=1

Matrix Arithmetic. j=1 An m n matrix is an array A = Matrix Arithmetic a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn of real numbers a ij An m n matrix has m rows and n columns a ij is the entry in the i-th row and j-th column

More information

Review of Basic Concepts in Linear Algebra

Review of Basic Concepts in Linear Algebra Review of Basic Concepts in Linear Algebra Grady B Wright Department of Mathematics Boise State University September 7, 2017 Math 565 Linear Algebra Review September 7, 2017 1 / 40 Numerical Linear Algebra

More information

Matrices. Chapter Definitions and Notations

Matrices. Chapter Definitions and Notations Chapter 3 Matrices 3. Definitions and Notations Matrices are yet another mathematical object. Learning about matrices means learning what they are, how they are represented, the types of operations which

More information

1 Last time: determinants

1 Last time: determinants 1 Last time: determinants Let n be a positive integer If A is an n n matrix, then its determinant is the number det A = Π(X, A)( 1) inv(x) X S n where S n is the set of n n permutation matrices Π(X, A)

More information

MATRICES. a m,1 a m,n A =

MATRICES. a m,1 a m,n A = MATRICES Matrices are rectangular arrays of real or complex numbers With them, we define arithmetic operations that are generalizations of those for real and complex numbers The general form a matrix of

More information

Lecture 3 Linear Algebra Background

Lecture 3 Linear Algebra Background Lecture 3 Linear Algebra Background Dan Sheldon September 17, 2012 Motivation Preview of next class: y (1) w 0 + w 1 x (1) 1 + w 2 x (1) 2 +... + w d x (1) d y (2) w 0 + w 1 x (2) 1 + w 2 x (2) 2 +...

More information

Calculating determinants for larger matrices

Calculating determinants for larger matrices Day 26 Calculating determinants for larger matrices We now proceed to define det A for n n matrices A As before, we are looking for a function of A that satisfies the product formula det(ab) = det A det

More information

Linear Algebra (Review) Volker Tresp 2017

Linear Algebra (Review) Volker Tresp 2017 Linear Algebra (Review) Volker Tresp 2017 1 Vectors k is a scalar (a number) c is a column vector. Thus in two dimensions, c = ( c1 c 2 ) (Advanced: More precisely, a vector is defined in a vector space.

More information

Review 1 Math 321: Linear Algebra Spring 2010

Review 1 Math 321: Linear Algebra Spring 2010 Department of Mathematics and Statistics University of New Mexico Review 1 Math 321: Linear Algebra Spring 2010 This is a review for Midterm 1 that will be on Thursday March 11th, 2010. The main topics

More information

Matrices BUSINESS MATHEMATICS

Matrices BUSINESS MATHEMATICS Matrices BUSINESS MATHEMATICS 1 CONTENTS Matrices Special matrices Operations with matrices Matrix multipication More operations with matrices Matrix transposition Symmetric matrices Old exam question

More information

10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections )

10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections ) c Dr. Igor Zelenko, Fall 2017 1 10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections 7.2-7.4) 1. When each of the functions F 1, F 2,..., F n in right-hand side

More information

LINEAR ALGEBRA REVIEW

LINEAR ALGEBRA REVIEW LINEAR ALGEBRA REVIEW When we define a term, we put it in boldface. This is a very compressed review; please read it very carefully and be sure to ask questions on parts you aren t sure of. x 1 WedenotethesetofrealnumbersbyR.

More information

DS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.

DS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra. DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1

More information

1 Matrices and matrix algebra

1 Matrices and matrix algebra 1 Matrices and matrix algebra 1.1 Examples of matrices A matrix is a rectangular array of numbers and/or variables. For instance 4 2 0 3 1 A = 5 1.2 0.7 x 3 π 3 4 6 27 is a matrix with 3 rows and 5 columns

More information

Lecture 2j Inner Product Spaces (pages )

Lecture 2j Inner Product Spaces (pages ) Lecture 2j Inner Product Spaces (pages 348-350) So far, we have taken the essential properties of R n and used them to form general vector spaces, and now we have taken the essential properties of the

More information

Homework 5 M 373K Mark Lindberg and Travis Schedler

Homework 5 M 373K Mark Lindberg and Travis Schedler Homework 5 M 373K Mark Lindberg and Travis Schedler 1. Artin, Chapter 3, Exercise.1. Prove that the numbers of the form a + b, where a and b are rational numbers, form a subfield of C. Let F be the numbers

More information

Matrices Gaussian elimination Determinants. Graphics 2009/2010, period 1. Lecture 4: matrices

Matrices Gaussian elimination Determinants. Graphics 2009/2010, period 1. Lecture 4: matrices Graphics 2009/2010, period 1 Lecture 4 Matrices m n matrices Matrices Definitions Diagonal, Identity, and zero matrices Addition Multiplication Transpose and inverse The system of m linear equations in

More information

MATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (in-class portion) January 22, 2003

MATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (in-class portion) January 22, 2003 MATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (in-class portion) January 22, 2003 1. True or False (28 points, 2 each) T or F If V is a vector space

More information

(K + L)(c x) = K(c x) + L(c x) (def of K + L) = K( x) + K( y) + L( x) + L( y) (K, L are linear) = (K L)( x) + (K L)( y).

(K + L)(c x) = K(c x) + L(c x) (def of K + L) = K( x) + K( y) + L( x) + L( y) (K, L are linear) = (K L)( x) + (K L)( y). Exercise 71 We have L( x) = x 1 L( v 1 ) + x 2 L( v 2 ) + + x n L( v n ) n = x i (a 1i w 1 + a 2i w 2 + + a mi w m ) i=1 ( n ) ( n ) ( n ) = x i a 1i w 1 + x i a 2i w 2 + + x i a mi w m i=1 Therefore y

More information

Math 360 Linear Algebra Fall Class Notes. a a a a a a. a a a

Math 360 Linear Algebra Fall Class Notes. a a a a a a. a a a Math 360 Linear Algebra Fall 2008 9-10-08 Class Notes Matrices As we have already seen, a matrix is a rectangular array of numbers. If a matrix A has m columns and n rows, we say that its dimensions are

More information

Review of Linear Algebra

Review of Linear Algebra Review of Linear Algebra Throughout these notes, F denotes a field (often called the scalars in this context). 1 Definition of a vector space Definition 1.1. A F -vector space or simply a vector space

More information

Linear Algebra (Review) Volker Tresp 2018

Linear Algebra (Review) Volker Tresp 2018 Linear Algebra (Review) Volker Tresp 2018 1 Vectors k, M, N are scalars A one-dimensional array c is a column vector. Thus in two dimensions, ( ) c1 c = c 2 c i is the i-th component of c c T = (c 1, c

More information

LINEAR ALGEBRA REVIEW

LINEAR ALGEBRA REVIEW LINEAR ALGEBRA REVIEW JC Stuff you should know for the exam. 1. Basics on vector spaces (1) F n is the set of all n-tuples (a 1,... a n ) with a i F. It forms a VS with the operations of + and scalar multiplication

More information

Phys 201. Matrices and Determinants

Phys 201. Matrices and Determinants Phys 201 Matrices and Determinants 1 1.1 Matrices 1.2 Operations of matrices 1.3 Types of matrices 1.4 Properties of matrices 1.5 Determinants 1.6 Inverse of a 3 3 matrix 2 1.1 Matrices A 2 3 7 =! " 1

More information

Materials engineering Collage \\ Ceramic & construction materials department Numerical Analysis \\Third stage by \\ Dalya Hekmat

Materials engineering Collage \\ Ceramic & construction materials department Numerical Analysis \\Third stage by \\ Dalya Hekmat Materials engineering Collage \\ Ceramic & construction materials department Numerical Analysis \\Third stage by \\ Dalya Hekmat Linear Algebra Lecture 2 1.3.7 Matrix Matrix multiplication using Falk s

More information

MATH 583A REVIEW SESSION #1

MATH 583A REVIEW SESSION #1 MATH 583A REVIEW SESSION #1 BOJAN DURICKOVIC 1. Vector Spaces Very quick review of the basic linear algebra concepts (see any linear algebra textbook): (finite dimensional) vector space (or linear space),

More information

Chapter Two Elements of Linear Algebra

Chapter Two Elements of Linear Algebra Chapter Two Elements of Linear Algebra Previously, in chapter one, we have considered single first order differential equations involving a single unknown function. In the next chapter we will begin to

More information

Review of Linear Algebra Definitions, Change of Basis, Trace, Spectral Theorem

Review of Linear Algebra Definitions, Change of Basis, Trace, Spectral Theorem Review of Linear Algebra Definitions, Change of Basis, Trace, Spectral Theorem Steven J. Miller June 19, 2004 Abstract Matrices can be thought of as rectangular (often square) arrays of numbers, or as

More information

Proofs for Quizzes. Proof. Suppose T is a linear transformation, and let A be a matrix such that T (x) = Ax for all x R m. Then

Proofs for Quizzes. Proof. Suppose T is a linear transformation, and let A be a matrix such that T (x) = Ax for all x R m. Then Proofs for Quizzes 1 Linear Equations 2 Linear Transformations Theorem 1 (2.1.3, linearity criterion). A function T : R m R n is a linear transformation if and only if a) T (v + w) = T (v) + T (w), for

More information

MATH 235. Final ANSWERS May 5, 2015

MATH 235. Final ANSWERS May 5, 2015 MATH 235 Final ANSWERS May 5, 25. ( points) Fix positive integers m, n and consider the vector space V of all m n matrices with entries in the real numbers R. (a) Find the dimension of V and prove your

More information

1 Matrices and Systems of Linear Equations. a 1n a 2n

1 Matrices and Systems of Linear Equations. a 1n a 2n March 31, 2013 16-1 16. Systems of Linear Equations 1 Matrices and Systems of Linear Equations An m n matrix is an array A = (a ij ) of the form a 11 a 21 a m1 a 1n a 2n... a mn where each a ij is a real

More information

MATH2210 Notebook 2 Spring 2018

MATH2210 Notebook 2 Spring 2018 MATH2210 Notebook 2 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 2 MATH2210 Notebook 2 3 2.1 Matrices and Their Operations................................

More information

Matrix representation of a linear map

Matrix representation of a linear map Matrix representation of a linear map As before, let e i = (0,..., 0, 1, 0,..., 0) T, with 1 in the i th place and 0 elsewhere, be standard basis vectors. Given linear map f : R n R m we get n column vectors

More information

MATH 240 Spring, Chapter 1: Linear Equations and Matrices

MATH 240 Spring, Chapter 1: Linear Equations and Matrices MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear

More information

Linear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.

Linear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0. Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the

More information

Review of some mathematical tools

Review of some mathematical tools MATHEMATICAL FOUNDATIONS OF SIGNAL PROCESSING Fall 2016 Benjamín Béjar Haro, Mihailo Kolundžija, Reza Parhizkar, Adam Scholefield Teaching assistants: Golnoosh Elhami, Hanjie Pan Review of some mathematical

More information

Mobile Robotics 1. A Compact Course on Linear Algebra. Giorgio Grisetti

Mobile Robotics 1. A Compact Course on Linear Algebra. Giorgio Grisetti Mobile Robotics 1 A Compact Course on Linear Algebra Giorgio Grisetti SA-1 Vectors Arrays of numbers They represent a point in a n dimensional space 2 Vectors: Scalar Product Scalar-Vector Product Changes

More information

Review problems for MA 54, Fall 2004.

Review problems for MA 54, Fall 2004. Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on

More information

Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form

Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1

More information

M. Matrices and Linear Algebra

M. Matrices and Linear Algebra M. Matrices and Linear Algebra. Matrix algebra. In section D we calculated the determinants of square arrays of numbers. Such arrays are important in mathematics and its applications; they are called matrices.

More information

INNER PRODUCT SPACE. Definition 1

INNER PRODUCT SPACE. Definition 1 INNER PRODUCT SPACE Definition 1 Suppose u, v and w are all vectors in vector space V and c is any scalar. An inner product space on the vectors space V is a function that associates with each pair of

More information

Linear Algebra Review

Linear Algebra Review Chapter 1 Linear Algebra Review It is assumed that you have had a course in linear algebra, and are familiar with matrix multiplication, eigenvectors, etc. I will review some of these terms here, but quite

More information

Some notes on linear algebra

Some notes on linear algebra Some notes on linear algebra Throughout these notes, k denotes a field (often called the scalars in this context). Recall that this means that there are two binary operations on k, denoted + and, that

More information

Final Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2

Final Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2 Final Review Sheet The final will cover Sections Chapters 1,2,3 and 4, as well as sections 5.1-5.4, 6.1-6.2 and 7.1-7.3 from chapters 5,6 and 7. This is essentially all material covered this term. Watch

More information

Matrix Basic Concepts

Matrix Basic Concepts Matrix Basic Concepts Topics: What is a matrix? Matrix terminology Elements or entries Diagonal entries Address/location of entries Rows and columns Size of a matrix A column matrix; vectors Special types

More information

Image Registration Lecture 2: Vectors and Matrices

Image Registration Lecture 2: Vectors and Matrices Image Registration Lecture 2: Vectors and Matrices Prof. Charlene Tsai Lecture Overview Vectors Matrices Basics Orthogonal matrices Singular Value Decomposition (SVD) 2 1 Preliminary Comments Some of this

More information

LECTURES 14/15: LINEAR INDEPENDENCE AND BASES

LECTURES 14/15: LINEAR INDEPENDENCE AND BASES LECTURES 14/15: LINEAR INDEPENDENCE AND BASES MA1111: LINEAR ALGEBRA I, MICHAELMAS 2016 1. Linear Independence We have seen in examples of span sets of vectors that sometimes adding additional vectors

More information

Systems of Linear Equations

Systems of Linear Equations Systems of Linear Equations Math 108A: August 21, 2008 John Douglas Moore Our goal in these notes is to explain a few facts regarding linear systems of equations not included in the first few chapters

More information

n n matrices The system of m linear equations in n variables x 1, x 2,..., x n can be written as a matrix equation by Ax = b, or in full

n n matrices The system of m linear equations in n variables x 1, x 2,..., x n can be written as a matrix equation by Ax = b, or in full n n matrices Matrices Definitions Diagonal, Identity, and zero matrices Addition Multiplication Transpose and inverse The system of m linear equations in n variables x 1, x 2,..., x n a 11 x 1 + a 12 x

More information

Homework Set #8 Solutions

Homework Set #8 Solutions Exercises.2 (p. 19) Homework Set #8 Solutions Assignment: Do #6, 8, 12, 14, 2, 24, 26, 29, 0, 2, 4, 5, 6, 9, 40, 42 6. Reducing the matrix to echelon form: 1 5 2 1 R2 R2 R1 1 5 0 18 12 2 1 R R 2R1 1 5

More information

Linear Algebra & Geometry why is linear algebra useful in computer vision?

Linear Algebra & Geometry why is linear algebra useful in computer vision? Linear Algebra & Geometry why is linear algebra useful in computer vision? References: -Any book on linear algebra! -[HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia

More information

Math 240 Calculus III

Math 240 Calculus III The Calculus III Summer 2015, Session II Wednesday, July 8, 2015 Agenda 1. of the determinant 2. determinants 3. of determinants What is the determinant? Yesterday: Ax = b has a unique solution when A

More information

TOPIC III LINEAR ALGEBRA

TOPIC III LINEAR ALGEBRA [1] Linear Equations TOPIC III LINEAR ALGEBRA (1) Case of Two Endogenous Variables 1) Linear vs. Nonlinear Equations Linear equation: ax + by = c, where a, b and c are constants. 2 Nonlinear equation:

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

More information

Introduction to Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Diego Tipaldi, Luciano Spinello

Introduction to Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Diego Tipaldi, Luciano Spinello Introduction to Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Diego Tipaldi, Luciano Spinello Vectors Arrays of numbers Vectors represent a point

More information

MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam. Topics

MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam. Topics MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam This study sheet will not be allowed during the test Books and notes will not be allowed during the test Calculators and cell phones

More information

Properties of Matrices and Operations on Matrices

Properties of Matrices and Operations on Matrices Properties of Matrices and Operations on Matrices A common data structure for statistical analysis is a rectangular array or matris. Rows represent individual observational units, or just observations,

More information

Elementary Row Operations on Matrices

Elementary Row Operations on Matrices King Saud University September 17, 018 Table of contents 1 Definition A real matrix is a rectangular array whose entries are real numbers. These numbers are organized on rows and columns. An m n matrix

More information

Math113: Linear Algebra. Beifang Chen

Math113: Linear Algebra. Beifang Chen Math3: Linear Algebra Beifang Chen Spring 26 Contents Systems of Linear Equations 3 Systems of Linear Equations 3 Linear Systems 3 2 Geometric Interpretation 3 3 Matrices of Linear Systems 4 4 Elementary

More information

Signature. Printed Name. Math 312 Hour Exam 1 Jerry L. Kazdan March 5, :00 1:20

Signature. Printed Name. Math 312 Hour Exam 1 Jerry L. Kazdan March 5, :00 1:20 Signature Printed Name Math 312 Hour Exam 1 Jerry L. Kazdan March 5, 1998 12:00 1:20 Directions: This exam has three parts. Part A has 4 True-False questions, Part B has 3 short answer questions, and Part

More information

Linear Algebra M1 - FIB. Contents: 5. Matrices, systems of linear equations and determinants 6. Vector space 7. Linear maps 8.

Linear Algebra M1 - FIB. Contents: 5. Matrices, systems of linear equations and determinants 6. Vector space 7. Linear maps 8. Linear Algebra M1 - FIB Contents: 5 Matrices, systems of linear equations and determinants 6 Vector space 7 Linear maps 8 Diagonalization Anna de Mier Montserrat Maureso Dept Matemàtica Aplicada II Translation:

More information

MATH 323 Linear Algebra Lecture 6: Matrix algebra (continued). Determinants.

MATH 323 Linear Algebra Lecture 6: Matrix algebra (continued). Determinants. MATH 323 Linear Algebra Lecture 6: Matrix algebra (continued). Determinants. Elementary matrices Theorem 1 Any elementary row operation σ on matrices with n rows can be simulated as left multiplication

More information

Matrices and Vectors

Matrices and Vectors Matrices and Vectors James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 11, 2013 Outline 1 Matrices and Vectors 2 Vector Details 3 Matrix

More information

Linear Algebra. Min Yan

Linear Algebra. Min Yan Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................

More information