Don t forget to think of a matrix as a map (a.k.a. A Linear Algebra Primer)

Size: px
Start display at page:

Download "Don t forget to think of a matrix as a map (a.k.a. A Linear Algebra Primer)"

Transcription

1 Don t forget to think of a matrix as a map (aka A Linear Algebra Primer) Thomas Yu February 5, Matrix Recall that a matrix is an array of numbers: a 11 a 1n A a m1 a mn In this note, we focus on the case of m n 2, although everything in this section has a natural generalization to general m and n (the concept of determinant, however, is only meaningful in the case of m n) It is important to not just think of a matrix as a boring array of number Matrix originated from the study of linear equations: a 11 x 1 + a 12 x a 1n x n b 1 a m1 x 1 + a m2 x a mn x n b m The first use of matrix is to write the above in the following compact form where A is a matrix, x and b are column vectors Ax b, The key point of this note is to urge you to think of a matrix as a function (aka a mapping/map) The study of various properties of linear solution urged mathematicians to think about the mapping: x [x 1, x 2 ] T Ax [a 11 x 1 + a 12 x 2, a 21 x 1 + a 22 x 2 ] T (11) As such, we think of a 2 2 matrix as a function from R 2 to R 2 (ie a function that maps two numbers to two numbers) Of course, there are many, many functions from R 2 to R 2 that are not of the form (11), those functions are called nonlinear functions The functions that are of the form (11) are called linear functions Linear functions are very special, and many basic properties of a linear function (eg 1-1, onto properties) can be determined easily The properties of linear maps are what you learn in a course of linear algebra You may wonder: why spend so much effort just to learn about one kind of functions? Besides the fact that there is much to learn about linear maps, an important point is that any nonlinear function can almost always be locally approximated by a linear function: if f : R 2 R 2 is differentiable near a point x R 2, then f1 f(x) f(x ) + x 1 (x f ) 1 x 2 (x ) f 2 x 1 (x f ) 2 x 2 (x (x x ), when x is near x ) 1

2 This has many important applications Once you start thinking matrices as functions, standard concepts pertaining to functions (function composition, 1-1, onto properties, etc etc) apply to matrices Composing two functions is something very basic, so one may ask: If we have two 2 2 matrices A and B, and we view them as functions from R 2 to R 2, then these functions can be composed Is the composed function still linear? If so, what is the new 2 2 matrix that represents the composite function? I put the answers to these two questions in the form of a theorem: Theorem 11 The composition of two linear functions is still linear The matrix that represents the composite function is exactly the matrix obtained by multiplying the two matrices in the way you were taught Proof: For clarity, we use the symbol f A : R 2 R 2 to denote the linear function represented by the matrix A, so if x R 2, f A (x) Ax [a 11 x 1 + a 12 x 2, a 21 x 1 + a 22 x 2 ] T Similarly we use the notation f B : R 2 R 2 to denote the linear function represented by the matrix B, so f B (x) Bx We seek to prove that the composite function is also linear Notice that f B f A : R 2 R 2 f B f A (x) f B (f A (x)) (def of function composition) B(Ax) a11 x B 1 + a 12 x 2 a 21 x 1 + a 22 x 2 b11 (a 11 x 1 + a 12 x 2 ) + b 12 (a 21 x 1 + a 22 x 2 ) b 21 (a 11 x 1 + a 12 x 2 ) + b 22 (a 21 x 1 + a 22 x 2 ) (b11 a 11 + b 12 a 21 )x 1 + (b 11 a 12 + b 12 a 22 )x 2 (b 21 a 11 + b 22 a 21 )x 1 + (b 21 a 12 + b 22 a 22 )x 2 b11 a 11 + b 12 a 21 b 11 a 12 + b 12 a 22 x1 b 21 a 11 + b 22 a 21 b 21 a 12 + b 22 a 22 x 2 So f B f A (x) is indeed a linear function and is represented by the matrix b11 a 11 + b 12 a 21 b 11 a 12 + b 12 a 22 ; b 21 a 11 + b 22 a 21 b 21 a 12 + b 22 a 22 moreover this matrix is exactly BA under the standard definition of matrix multiplication The main point here is that the definition of matrix multiplication which may look quite unmotivated as first glance is exactly the operation for composing two linear maps Given the above interpretation of matrix multiplication, it is hardly surprising that matrix multiplication is not commutative, ie AB is not always equal to BA Composing two maps in two different orders typically lead to very different composite maps (A non-mathematical example: Compare (i) you put clothes into a washing machine, take them out, and then put them into a dryer; versus (ii) you put clothes into a drying machine, take them out, and then put them into a washing machine You get totally different results!) Exercise: Consider the square Let A 2 3, B Calculate P AB, Q BA S {[x, y] T : x 1, y 1} Now, determine the 4 corners of each of the following parallelograms A(S), B(S), P (S), Q(S), and draw these parallelograms 2

3 (Again, we view a matrix as a function from R 2 to R 2, so if A is a matrix and S R 2, A(S) is the image of the set S under the function A as defined in the book, ie A(S) {Ax : x S}, which is also a subset of R 2 in this case) Notice that P (S) and Q(S) are two different parallelograms; compare their areas 2 Determinant So now you understand that we should think of a matrix as a transformation on R 2 Different matrices have different actions on R 2 Indeed, some matrices can transform the plane quite radically, as illustrated by 1 2 A (22) 1 2 This matrix maps every point on the plane onto the line {[x, y] T : x y} (see it?), it collapses any region with a positive area into a region with zero area The determinant of a 2 2 matrix is given by the following simple formula: a11 a det( 12 ) a a 21 a 11 a 22 a 12 a The simplicity of this formula is deceptive Where does this formula come from? What does this magical number say about the matrix itself? Below I give the answers to these questions, with proofs omitted Let S R 2, f : R 2 R 2, and I want to measure: area(f(s)) area(s) Here, for a technical reason, I have to assume that S is nice in a certain sense 1 in order to talk about its area, a measurable set S can be quite complicated and needs not be of a familiar shape such as a polygon or an ellipse In general, the above ratio depends on both the function f and the subset S: a general nonlinear function f : R 2 R 2 can distort area mildly at some places but very wildly at some other places However, a linear function cannot do that; we have: Theorem 21 (I) The above ratio does not depend on S when f is a linear mapping, ie a function of the form (11) (II) So the ratio is only dependent on the matrix A and this ratio is exactly det(a) I will skip the proof of the above important fact For one thing, I did not clarify what I mean by S being nice, and when S is nice, how do I exactly define the area of S However, the above theorem is not too hard to prove if I assume S to be just rectangles; and it is a very good exercise to do Now I have explained what the magnitude of a matrix measures about the matrix itself What, then, does the sign of the determinant of a matrix say about the matrix? It is not hard to see that a matrix preserves orientation if and only if its determinant is positive; equivalently, a matrix reverses orientation if and only if its determinant is negative You may remember learning about the following fact: Theorem 22 A is invertible if and only if det(a) 0 The above theorem is actually closely related to Theorem 21 (although here I am not attempting to prove either one of the two theorems using the other): Recall that any function is invertible if and only if 1 The technical term for niceness here is measurable, a concept developed in a subject called measure theory 3

4 it is 1-1 and onto In general, 1-1 and onto are two totally different concepts; however, any linear function from R n to R n is 1-1 if and only if it is onto So a matrix A is invertible iff the associated linear map is onto When is a function onto? Answer: exactly when the linear map does not collapse any subset with a positive area into a subset with zero area, ie when the determinant is not zero For example, the matrix in (22) has a zero determinant, it is non-invertible, and, as a map, it collapses any region with a positive area to a set with zero area Theorem 23 det(ab) det(a) det(b) det(ba) (although in general AB BA) Proof: Let S be any nice subset of R 2 with a positive area, then det(ab) T hm21 area(ab(s)) area(s) area(a(b(s))) area(b(s)) T hm21 det(a) det(b) area(b(s)) area(s) 3 Orthogonal matrices and matrices with determinant ±1 To wrap up this crash course on linear algebra, we compare here matrices which (i) are orthogonal (Definition: an n n matrix is orthogonal if A 1 A T ) and (ii) have a determinant equals to 1 or 1 Anyone seeing the condition A 1 A T the first time must ask: what does this condition really say about the linear map defined by the matrix A? The concept of orthogonal matrices come from the following fact: Theorem 31 A linear map preserves Euclidean distance if and only if it is one represented by an orthogonal matrix (As such, an orthogonal matrix represents a linear map that is like a rotation, a reflection, or a combination of the two) Proof: ( ) Recall that the Euclidean distance x of a vector x R n is: x n x 2 i x T x i1 Therefore if A T A I, then Ax 2 (Ax) T (Ax) x T A T Ax x T (I)x x T x x 2 In other words, the the action of A to any vector can not change the length of the vector ( ) (The proof of this part can be skipped) First of all, notice that if A preserves length, it must be invertible: assume not, then a very basic result in linear algebra says that Ax 0 for some non-zero vector x; but this immediately violates the length preserving assumption Let r 1,, r n be the rows of the n n matrix A, then r1 T,, r n are column vectors Note that no row is identically zero (otherwise A is singular), so ri T > 0 for all i Next, consider r 1 r T r1 T 1 Ar1 T r 2 r T 1 r n r1 T r n r1 T Since A preserves length, Ar1 T r1 T ; this also means r 2 r1 T r n r1 T 0 Similarly, we can show that r i rj T 0, i j So all together we have shown that all the off-diagonal entries of AA T are zeros, and all the diagonal entries of AA T are positive This also means that the inverse of AA T has the same structure 4

5 To finish the proof, we need to show that the diagonal entries of (AA T ) 1 are all equal to one Note that A 1 also preserves length: A 1 y A(A 1 y) y for any y So y T A T A 1 y y T y for any y By choosing y e i (ie the column vector with zeros in all but the ith position, and equals 1 at the ith position), we have (A T A 1 ) i,i e T i (A T A 1 )e i e T i e i 1 Note: the ( ) part of the above theorem is the same as Property 732 on Page 398 of DeGroot and Schervish Next we want to show that a an orthogonal matrix must have determinant equals to 1 The proof of this relies on the algebraic fact that det(a T ) det(a) for any square matrix A: 1 det(i) det(a T A) det(a T ) det(a) det(a) 2 det(a) ±1 By our earlier interpretation of the determinant (Theorem 21), the above results mean that a length preserving linear map must also preserve area/volume/measure (when n 2/n 3/general n) You may now wonder: is it true that any (say 2 2) area preserving matrix must also preserve length? The answer is negative, consider the following shearing map: 1 1 A 0 1 It has determinant equals to 1, meaning that it preserves area In fact, A shears any square into a parallelogram of the same area (Why? Please draw[ some [ pictures) ] [ However, A T A I, and a shearing map clearly does not preserve length; for instance, 0 1] 1 1] 4 Conclusion Many concepts in linear algebra which at first glance look rather un-motivating have simple and elegant geometric interpretations The key is not to forget that a matrix should be interpreted as a linear map 5

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

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

Honors Advanced Mathematics Determinants page 1

Honors Advanced Mathematics Determinants page 1 Determinants page 1 Determinants For every square matrix A, there is a number called the determinant of the matrix, denoted as det(a) or A. Sometimes the bars are written just around the numbers of the

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

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

Vector Spaces. Addition : R n R n R n Scalar multiplication : R R n R n.

Vector Spaces. Addition : R n R n R n Scalar multiplication : R R n R n. Vector Spaces Definition: The usual addition and scalar multiplication of n-tuples x = (x 1,..., x n ) R n (also called vectors) are the addition and scalar multiplication operations defined component-wise:

More information

Math 313 (Linear Algebra) Exam 2 - Practice Exam

Math 313 (Linear Algebra) Exam 2 - Practice Exam Name: Student ID: Section: Instructor: Math 313 (Linear Algebra) Exam 2 - Practice Exam Instructions: For questions which require a written answer, show all your work. Full credit will be given only if

More information

1 Matrices and Systems of Linear Equations

1 Matrices and Systems of Linear Equations March 3, 203 6-6. Systems of Linear Equations Matrices and Systems of Linear Equations An m n matrix is an array A = a ij of the form a a n a 2 a 2n... a m a mn where each a ij is a real or complex number.

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

12. Perturbed Matrices

12. Perturbed Matrices MAT334 : Applied Linear Algebra Mike Newman, winter 208 2. Perturbed Matrices motivation We want to solve a system Ax = b in a context where A and b are not known exactly. There might be experimental errors,

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

Note: Every graph is a level set (why?). But not every level set is a graph. Graphs must pass the vertical line test. (Level sets may or may not.

Note: Every graph is a level set (why?). But not every level set is a graph. Graphs must pass the vertical line test. (Level sets may or may not. Curves in R : Graphs vs Level Sets Graphs (y = f(x)): The graph of f : R R is {(x, y) R y = f(x)} Example: When we say the curve y = x, we really mean: The graph of the function f(x) = x That is, we mean

More information

Review of linear algebra

Review of linear algebra 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

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

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

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

MAT 1332: CALCULUS FOR LIFE SCIENCES. Contents. 1. Review: Linear Algebra II Vectors and matrices Definition. 1.2.

MAT 1332: CALCULUS FOR LIFE SCIENCES. Contents. 1. Review: Linear Algebra II Vectors and matrices Definition. 1.2. MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI Contents 1 Review: Linear Algebra II Vectors and matrices 1 11 Definition 1 12 Operations 1 2 Linear Algebra III Inverses and Determinants 1 21 Inverse Matrices

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

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

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

Math 416, Spring 2010 The algebra of determinants March 16, 2010 THE ALGEBRA OF DETERMINANTS. 1. Determinants

Math 416, Spring 2010 The algebra of determinants March 16, 2010 THE ALGEBRA OF DETERMINANTS. 1. Determinants THE ALGEBRA OF DETERMINANTS 1. Determinants We have already defined the determinant of a 2 2 matrix: det = ad bc. We ve also seen that it s handy for determining when a matrix is invertible, and when it

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

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

DETERMINANTS DEFINED BY ROW OPERATIONS

DETERMINANTS DEFINED BY ROW OPERATIONS DETERMINANTS DEFINED BY ROW OPERATIONS TERRY A. LORING. DETERMINANTS DEFINED BY ROW OPERATIONS Determinants of square matrices are best understood in terms of row operations, in my opinion. Most books

More information

Chapter 8. Rigid transformations

Chapter 8. Rigid transformations Chapter 8. Rigid transformations We are about to start drawing figures in 3D. There are no built-in routines for this purpose in PostScript, and we shall have to start more or less from scratch in extending

More information

1 0 1, then use that decomposition to solve the least squares problem. 1 Ax = 2. q 1 = a 1 a 1 = 1. to find the intermediate result:

1 0 1, then use that decomposition to solve the least squares problem. 1 Ax = 2. q 1 = a 1 a 1 = 1. to find the intermediate result: Exercise Find the QR decomposition of A =, then use that decomposition to solve the least squares problem Ax = 2 3 4 Solution Name the columns of A by A = [a a 2 a 3 ] and denote the columns of the results

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

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

2018 Fall 2210Q Section 013 Midterm Exam II Solution

2018 Fall 2210Q Section 013 Midterm Exam II Solution 08 Fall 0Q Section 0 Midterm Exam II Solution True or False questions points 0 0 points) ) Let A be an n n matrix. If the equation Ax b has at least one solution for each b R n, then the solution is unique

More information

CS 143 Linear Algebra Review

CS 143 Linear Algebra Review CS 143 Linear Algebra Review Stefan Roth September 29, 2003 Introductory Remarks This review does not aim at mathematical rigor very much, but instead at ease of understanding and conciseness. Please see

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

3 Fields, Elementary Matrices and Calculating Inverses

3 Fields, Elementary Matrices and Calculating Inverses 3 Fields, Elementary Matrices and Calculating Inverses 3. Fields So far we have worked with matrices whose entries are real numbers (and systems of equations whose coefficients and solutions are real numbers).

More information

Examples True or false: 3. Let A be a 3 3 matrix. Then there is a pattern in A with precisely 4 inversions.

Examples True or false: 3. Let A be a 3 3 matrix. Then there is a pattern in A with precisely 4 inversions. The exam will cover Sections 6.-6.2 and 7.-7.4: True/False 30% Definitions 0% Computational 60% Skip Minors and Laplace Expansion in Section 6.2 and p. 304 (trajectories and phase portraits) in Section

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

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

Definition 2.3. We define addition and multiplication of matrices as follows.

Definition 2.3. We define addition and multiplication of matrices as follows. 14 Chapter 2 Matrices In this chapter, we review matrix algebra from Linear Algebra I, consider row and column operations on matrices, and define the rank of a matrix. Along the way prove that the row

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

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

Evaluating Determinants by Row Reduction

Evaluating Determinants by Row Reduction Evaluating Determinants by Row Reduction MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Objectives Reduce a matrix to row echelon form and evaluate its determinant.

More information

Linear Algebra, Summer 2011, pt. 2

Linear Algebra, Summer 2011, pt. 2 Linear Algebra, Summer 2, pt. 2 June 8, 2 Contents Inverses. 2 Vector Spaces. 3 2. Examples of vector spaces..................... 3 2.2 The column space......................... 6 2.3 The null space...........................

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

1300 Linear Algebra and Vector Geometry

1300 Linear Algebra and Vector Geometry 1300 Linear Algebra and Vector Geometry R. Craigen Office: MH 523 Email: craigenr@umanitoba.ca May-June 2017 Matrix Inversion Algorithm One payoff from this theorem: It gives us a way to invert matrices.

More information

Equality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.

Equality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same. Introduction Matrix Operations Matrix: An m n matrix A is an m-by-n array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read

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

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

LINEAR ALGEBRA KNOWLEDGE SURVEY

LINEAR ALGEBRA KNOWLEDGE SURVEY LINEAR ALGEBRA KNOWLEDGE SURVEY Instructions: This is a Knowledge Survey. For this assignment, I am only interested in your level of confidence about your ability to do the tasks on the following pages.

More information

LINEAR ALGEBRA REVIEW

LINEAR ALGEBRA REVIEW LINEAR ALGEBRA REVIEW SPENCER BECKER-KAHN Basic Definitions Domain and Codomain. Let f : X Y be any function. This notation means that X is the domain of f and Y is the codomain of f. This means that for

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 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

Linear Algebra Primer

Linear Algebra Primer Introduction Linear Algebra Primer Daniel S. Stutts, Ph.D. Original Edition: 2/99 Current Edition: 4//4 This primer was written to provide a brief overview of the main concepts and methods in elementary

More information

Math Lecture 26 : The Properties of Determinants

Math Lecture 26 : The Properties of Determinants Math 2270 - Lecture 26 : The Properties of Determinants Dylan Zwick Fall 202 The lecture covers section 5. from the textbook. The determinant of a square matrix is a number that tells you quite a bit about

More information

SPRING OF 2008 D. DETERMINANTS

SPRING OF 2008 D. DETERMINANTS 18024 SPRING OF 2008 D DETERMINANTS In many applications of linear algebra to calculus and geometry, the concept of a determinant plays an important role This chapter studies the basic properties of determinants

More information

Linear Algebra Primer

Linear Algebra Primer Linear Algebra Primer D.S. Stutts November 8, 995 Introduction This primer was written to provide a brief overview of the main concepts and methods in elementary linear algebra. It was not intended to

More information

A = , A 32 = n ( 1) i +j a i j det(a i j). (1) j=1

A = , A 32 = n ( 1) i +j a i j det(a i j). (1) j=1 Lecture Notes: Determinant of a Square Matrix Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk 1 Determinant Definition Let A [a ij ] be an

More information

22A-2 SUMMER 2014 LECTURE Agenda

22A-2 SUMMER 2014 LECTURE Agenda 22A-2 SUMMER 204 LECTURE 2 NATHANIEL GALLUP The Dot Product Continued Matrices Group Work Vectors and Linear Equations Agenda 2 Dot Product Continued Angles between vectors Given two 2-dimensional vectors

More information

Computationally, diagonal matrices are the easiest to work with. With this idea in mind, we introduce similarity:

Computationally, diagonal matrices are the easiest to work with. With this idea in mind, we introduce similarity: Diagonalization We have seen that diagonal and triangular matrices are much easier to work with than are most matrices For example, determinants and eigenvalues are easy to compute, and multiplication

More information

Practice problems for Exam 3 A =

Practice problems for Exam 3 A = Practice problems for Exam 3. Let A = 2 (a) Determine whether A is diagonalizable. If so, find a matrix S such that S AS is diagonal. If not, explain why not. (b) What are the eigenvalues of A? Is A diagonalizable?

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

9.4 Radical Expressions

9.4 Radical Expressions Section 9.4 Radical Expressions 95 9.4 Radical Expressions In the previous two sections, we learned how to multiply and divide square roots. Specifically, we are now armed with the following two properties.

More information

8 Square matrices continued: Determinants

8 Square matrices continued: Determinants 8 Square matrices continued: Determinants 8.1 Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You

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

Chapter 2. Matrix Arithmetic. Chapter 2

Chapter 2. Matrix Arithmetic. Chapter 2 Matrix Arithmetic Matrix Addition and Subtraction Addition and subtraction act element-wise on matrices. In order for the addition/subtraction (A B) to be possible, the two matrices A and B must have the

More information

4. Determinants.

4. Determinants. 4. Determinants 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 2 2 determinant 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 3 3 determinant 4.1.

More information

Math 309 Notes and Homework for Days 4-6

Math 309 Notes and Homework for Days 4-6 Math 309 Notes and Homework for Days 4-6 Day 4 Read Section 1.2 and the notes below. The following is the main definition of the course. Definition. A vector space is a set V (whose elements are called

More information

MATRICES AND MATRIX OPERATIONS

MATRICES AND MATRIX OPERATIONS SIZE OF THE MATRIX is defined by number of rows and columns in the matrix. For the matrix that have m rows and n columns we say the size of the matrix is m x n. If matrix have the same number of rows (n)

More information

MAT224 Practice Exercises - Week 7.5 Fall Comments

MAT224 Practice Exercises - Week 7.5 Fall Comments MAT224 Practice Exercises - Week 75 Fall 27 Comments The purpose of this practice exercise set is to give a review of the midterm material via easy proof questions You can read it as a summary and you

More information

Linear Algebra (part 1) : Matrices and Systems of Linear Equations (by Evan Dummit, 2016, v. 2.02)

Linear Algebra (part 1) : Matrices and Systems of Linear Equations (by Evan Dummit, 2016, v. 2.02) Linear Algebra (part ) : Matrices and Systems of Linear Equations (by Evan Dummit, 206, v 202) Contents 2 Matrices and Systems of Linear Equations 2 Systems of Linear Equations 2 Elimination, Matrix Formulation

More information

Announcements Wednesday, October 25

Announcements Wednesday, October 25 Announcements Wednesday, October 25 The midterm will be returned in recitation on Friday. The grade breakdown is posted on Piazza. You can pick it up from me in office hours before then. Keep tabs on your

More information

Announcements Wednesday, November 01

Announcements Wednesday, November 01 Announcements Wednesday, November 01 WeBWorK 3.1, 3.2 are due today at 11:59pm. The quiz on Friday covers 3.1, 3.2. My office is Skiles 244. Rabinoffice hours are Monday, 1 3pm and Tuesday, 9 11am. Section

More information

A First Course in Linear Algebra

A First Course in Linear Algebra A First Course in Linear Algebra About the Author Mohammed Kaabar is a math tutor at the Math Learning Center (MLC) at Washington State University, Pullman, and he is interested in linear algebra, scientific

More information

Inner product spaces. Layers of structure:

Inner product spaces. Layers of structure: Inner product spaces Layers of structure: vector space normed linear space inner product space The abstract definition of an inner product, which we will see very shortly, is simple (and by itself is pretty

More information

SOLUTION KEY TO THE LINEAR ALGEBRA FINAL EXAM 1 2 ( 2) ( 1) c a = 1 0

SOLUTION KEY TO THE LINEAR ALGEBRA FINAL EXAM 1 2 ( 2) ( 1) c a = 1 0 SOLUTION KEY TO THE LINEAR ALGEBRA FINAL EXAM () We find a least squares solution to ( ) ( ) A x = y or 0 0 a b = c 4 0 0. 0 The normal equation is A T A x = A T y = y or 5 0 0 0 0 0 a b = 5 9. 0 0 4 7

More information

Inverses and Determinants

Inverses and Determinants Engineering Mathematics 1 Fall 017 Inverses and Determinants I begin finding the inverse of a matrix; namely 1 4 The inverse, if it exists, will be of the form where AA 1 I; which works out to ( 1 4 A

More information

Matrix operations Linear Algebra with Computer Science Application

Matrix operations Linear Algebra with Computer Science Application Linear Algebra with Computer Science Application February 14, 2018 1 Matrix operations 11 Matrix operations If A is an m n matrix that is, a matrix with m rows and n columns then the scalar entry in the

More information

February 20 Math 3260 sec. 56 Spring 2018

February 20 Math 3260 sec. 56 Spring 2018 February 20 Math 3260 sec. 56 Spring 2018 Section 2.2: Inverse of a Matrix Consider the scalar equation ax = b. Provided a 0, we can solve this explicity x = a 1 b where a 1 is the unique number such that

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

Topic 15 Notes Jeremy Orloff

Topic 15 Notes Jeremy Orloff Topic 5 Notes Jeremy Orloff 5 Transpose, Inverse, Determinant 5. Goals. Know the definition and be able to compute the inverse of any square matrix using row operations. 2. Know the properties of inverses.

More information

Undergraduate Mathematical Economics Lecture 1

Undergraduate Mathematical Economics Lecture 1 Undergraduate Mathematical Economics Lecture 1 Yu Ren WISE, Xiamen University September 15, 2014 Outline 1 Courses Description and Requirement 2 Course Outline ematical techniques used in economics courses

More information

Graduate Mathematical Economics Lecture 1

Graduate Mathematical Economics Lecture 1 Graduate Mathematical Economics Lecture 1 Yu Ren WISE, Xiamen University September 23, 2012 Outline 1 2 Course Outline ematical techniques used in graduate level economics courses Mathematics for Economists

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

Jim Lambers MAT 610 Summer Session Lecture 1 Notes

Jim Lambers MAT 610 Summer Session Lecture 1 Notes Jim Lambers MAT 60 Summer Session 2009-0 Lecture Notes Introduction This course is about numerical linear algebra, which is the study of the approximate solution of fundamental problems from linear algebra

More information

A Brief Outline of Math 355

A Brief Outline of Math 355 A Brief Outline of Math 355 Lecture 1 The geometry of linear equations; elimination with matrices A system of m linear equations with n unknowns can be thought of geometrically as m hyperplanes intersecting

More information

1. In this problem, if the statement is always true, circle T; otherwise, circle F.

1. In this problem, if the statement is always true, circle T; otherwise, circle F. Math 1553, Extra Practice for Midterm 3 (sections 45-65) Solutions 1 In this problem, if the statement is always true, circle T; otherwise, circle F a) T F If A is a square matrix and the homogeneous equation

More information

Determinants An Introduction

Determinants An Introduction Determinants An Introduction Professor Je rey Stuart Department of Mathematics Paci c Lutheran University Tacoma, WA 9844 USA je rey.stuart@plu.edu The determinant is a useful function that takes a square

More information

Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI?

Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI? Section 5. The Characteristic Polynomial Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of A-cI? Property The eigenvalues

More information

Transpose & Dot Product

Transpose & Dot Product Transpose & Dot Product Def: The transpose of an m n matrix A is the n m matrix A T whose columns are the rows of A. So: The columns of A T are the rows of A. The rows of A T are the columns of A. Example:

More information

NOTES ON VECTORS, PLANES, AND LINES

NOTES ON VECTORS, PLANES, AND LINES NOTES ON VECTORS, PLANES, AND LINES DAVID BEN MCREYNOLDS 1. Vectors I assume that the reader is familiar with the basic notion of a vector. The important feature of the vector is that it has a magnitude

More information

We could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2

We could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2 Week 22 Equations, Matrices and Transformations Coefficient Matrix and Vector Forms of a Linear System Suppose we have a system of m linear equations in n unknowns a 11 x 1 + a 12 x 2 + + a 1n x n b 1

More information

Introduction to Matrix Algebra

Introduction to Matrix Algebra Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p

More information

Determinants and Scalar Multiplication

Determinants and Scalar Multiplication Invertibility and Properties of Determinants In a previous section, we saw that the trace function, which calculates the sum of the diagonal entries of a square matrix, interacts nicely with the operations

More information

Announcements Wednesday, November 01

Announcements Wednesday, November 01 Announcements Wednesday, November 01 WeBWorK 3.1, 3.2 are due today at 11:59pm. The quiz on Friday covers 3.1, 3.2. My office is Skiles 244. Rabinoffice hours are Monday, 1 3pm and Tuesday, 9 11am. Section

More information

Section 4.5. Matrix Inverses

Section 4.5. Matrix Inverses Section 4.5 Matrix Inverses The Definition of Inverse Recall: The multiplicative inverse (or reciprocal) of a nonzero number a is the number b such that ab = 1. We define the inverse of a matrix in almost

More information

Math 308 Midterm Answers and Comments July 18, Part A. Short answer questions

Math 308 Midterm Answers and Comments July 18, Part A. Short answer questions Math 308 Midterm Answers and Comments July 18, 2011 Part A. Short answer questions (1) Compute the determinant of the matrix a 3 3 1 1 2. 1 a 3 The determinant is 2a 2 12. Comments: Everyone seemed to

More information

and let s calculate the image of some vectors under the transformation T.

and let s calculate the image of some vectors under the transformation T. Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =

More information

Matrices. Chapter Keywords and phrases. 3.2 Introduction

Matrices. Chapter Keywords and phrases. 3.2 Introduction Chapter 3 Matrices 3.1 Keywords and phrases Special matrices: (row vector, column vector, zero, square, diagonal, scalar, identity, lower triangular, upper triangular, symmetric, row echelon form, reduced

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

Determinants in detail

Determinants in detail Determinants in detail Kyle Miller September 27, 2016 The book s introduction to the determinant det A of an n n square matrix A is to say there is a quantity which determines exactly when A is invertible,

More information

CS 468: Computational Topology Group Theory Fall b c b a b a c b a c b c c b a

CS 468: Computational Topology Group Theory Fall b c b a b a c b a c b c c b a Q: What s purple and commutes? A: An abelian grape! Anonymous Group Theory Last lecture, we learned about a combinatorial method for characterizing spaces: using simplicial complexes as triangulations

More information

MTH 464: Computational Linear Algebra

MTH 464: Computational Linear Algebra MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University March 2, 2018 Linear Algebra (MTH 464)

More information

Chapter 5: Matrices. Daniel Chan. Semester UNSW. Daniel Chan (UNSW) Chapter 5: Matrices Semester / 33

Chapter 5: Matrices. Daniel Chan. Semester UNSW. Daniel Chan (UNSW) Chapter 5: Matrices Semester / 33 Chapter 5: Matrices Daniel Chan UNSW Semester 1 2018 Daniel Chan (UNSW) Chapter 5: Matrices Semester 1 2018 1 / 33 In this chapter Matrices were first introduced in the Chinese Nine Chapters on the Mathematical

More information