Linear Algebra Handout

Size: px
Start display at page:

Download "Linear Algebra Handout"

Transcription

1 Linear Algebra Handout References Some material and suggested problems are taken from Fundamentals of Matrix Algebra by Gregory Hartman, which can be found here: I ll reference this book as FMA in the rest of this handout. Matrix Addition and Multiplication You need to know how to do multiply a matrix by a scalar and how to add matrices. You should know how the dimensions need to match in order to multiply matrices (and what the dimensions of the resulting product will be) and how to multiply two matrices. See sections. and. of FMA if you need a review of these topics or practice problems. Elementary Row Operations Definition (Elementary Row [Column] Operations).. Multiply a row by a scalar and add it to another row (replacing the latter row with that sum).. Multiply a row by a nonzero scalar.. Interchange/swap two rows. [There are analogous definitions for elementary column operations.] To do these operations using matrix multiplication, we first create an elementary matrix by starting with an appropriately sized identity matrix and then performing that operation on it. Then we multiply our matrix A by that elementary matrix. To perform row operations we pre-multiply A by the elementary matrix, i.e., multiply it on the left: EA; to perform column operations we post-multiply A by the elementary matrix, i.e., multiply it on the right: AE.. Let A = (a) Create elementary matrices that perform the following operations on A: Swap rows and. Multiply row by -. Add times row to row. Zero out row [technically not an elementary row operation, but still the same idea].

2 Swap rows and. (b) Now perform all of those row operations on A in the order listed. (c) Now find a single matrix that will perform all of those row operations on A. (d) Create elementary matrices that perform the following operations on A: Add times column to column. Multiply column by. Zero out column [technically not an elementary column operation, but still the same idea]. Interchange columns and. (e) Now perform all of those column operations on A in the order listed. (f) Now find a single matrix that will perform all of those column operations on A.. Find the matrix of the elementary row operations which swaps the second and fifth rows of matrices with five rows. Find the square of this matrix and interpret the result in terms of elementary row operations. Now do the same for elementary column operations.. Find a sequence of matrices E, E, E, and E which reduce A = E A =, E E A =, E E E A =, and E E E E A = I. to the following forms: Reduced Row Echelon Form (Discussed in sections. and. of FMA) Definition (Reduced Row Echelon Form). A matrix is in reduced row echelon form (abbr. rref) if its entries satisfy the following conditions.. The first nonzero entry in each row is a (called a leading ).. Each leading comes in a column to the right of the leading s in rows above it.. All rows of all s come at the bottom of the matrix.. If a column contains a leading, then all other entries in that column are. A matrix that satisfies the first three conditions is said to be in row echelon form. Algorithm (Gaussian Elimination). Here is the algorithm (known as Gaussian Elimination ) for putting a matrix in rref: Forward Steps. Working from left to right, consider the first column that isn t all zeros that hasn t already been worked on. Then working from top to bottom, consider the first row that hasn t been worked on.. If the entry in the row and column that we are considering is zero, interchange rows with a row below the current row so that that entry is nonzero. If all entries below are zero, we are done with this column; start again at step.. Multiply the current row by a scalar to make its first entry a (a leading ).. Repeatedly use Elementary Row Operation to put zeros underneath the leading one. 5. Go back to step and work on the new rows and columns until either all rows or columns have been worked on.

3 If the above steps have been followed properly, then the following should be true about the current state of the matrix:. The first nonzero entry in each row is a (a leading ).. Each leading is in a column to the right of the leading s above it.. All rows of all zeros come at the bottom of the matrix. Note that this means we have just put a matrix into row echelon form. The next steps finish the conversion into reduced row echelon form. These next steps are referred to as the backward steps. These are much easier to state. Backward Steps. Starting from the right and working left, use Elementary Row Operation repeatedly to put zeros above each leading. Work through the following exercises from FMA: From section.:, 9, 7, 9,. From section.:, 9,,. Solving Matrix/Vector Equations (A x = b) and Matrix/Matrix Equations (A X = B) (See sections. and.5 of FMA.) To solve either of these types of equations, First form the augmented matrix by writing down the entries of A on one side of the augmented matrix (typically the left if doing this by hand, and the need to be on the left if doing the next step in Mathematica) and the entries of b or B on the right. Then put the augmented matrix into rref using elementary row operations. Read off the solution. If solving a matrix/vector equation, the independent/free variables correspond to columns in which there is no leading and the dependent/basic/bound variables correspond to columns in which there is a leading. We then write the basic variables in terms of the free variables and can further break up the solution into its particular part and the part corresponding to the associated homogeneous equation A x =. For example, if we ended up with an augmented matrix of the form 5 5 then x is free while x and x are basic. From the first row we see that x + 5x = so x = 5x, and from the second row x = 5. Thus the solution vector would look like 5x x = x 5 5 = + x. 5

4 5 There is the particular solution and any scalar multiple of would be a solution 5 to A x =. If solving a matrix/matrix equation, the solution is the right side of the augmented matrix as long as the left side where A was originally is now the identity matrix! (If not then there is no solution.) Note that when solving a matrix/vector equation there are three possible scenarios:. There is no solution. (This happens when you encounter a contradiction in the equations, which will look like having = something nonzero when you read the solution off of the rref augmented matrix.). There is one unique solution. (This will happen when the part of the augmented matrix corresponding to A becomes the identity in the rref.). There are infinitely many solutions. (This happens if there are free variables.). Work through the following exercises from FMA: From section.: 9,,. From section.5: 7, 9,.. For each of the following A and b solve the equation A x = b using Mathematica: (a) A =, b = 6 (b) A = 5, b =. For each of the following A and B solve the equation A X = B using Mathematica: (a) A =, B = (b) A = 5, B = 6 6

5 5 Span and Spanning Sets Given a set of vectors, we would like to know which other vectors we can create by adding scalar multiples of the vectors in the set. Definition. The span of a set of vectors V = { v, v,..., v n } is the set of all linear combinations of the vectors in V, i.e., all vectors of the form where α, α,..., α n are scalars (real numbers). x = α v + α v + + α n v n We denote the span of V by span(v ) or, if listing out the individual vectors, span{ v, v,..., v n } So how do we figure out which vectors are in the span of vectors v, v,..., v n? The general steps are:. Create a matrix by making v, v,..., v n the rows, maintaining the same relative position (i.e., v becomes row, v becomes row, and so on). This may seem unnatural since we typically write vectors in column form, but we re going to be performing elementary row operations on this matrix we ve created and making the vectors the rows will mean that we re performing those same operations on v, v,..., v n.. Now put this matrix into rref.. The span of the rows of the rref matrix will be the same as the span of the original set { v, v,..., v n } (we call this a basis for the span of those vectors). In other words, any vector that can be created from a linear combination of vectors from { v, v,..., v n } can also be created from a linear combination of the rows of the rref form of the matrix. If any of the rows are all s, then that tells us that the corresponding vectors in the original set are redundant and not necessary Example. Given the two sets of vectors V = u =, u =, u = 6 7 and V = w =, w = 5. Is span(v ) = span(v )? Let A be the matrix consisting of the V vectors and B be the matrix consisting of the V vectors. Putting these matrices into rref yields: / A = rref 8/ 6 7 and B = rref /. 5 8/ 5

6 The nonzero row of each rref are the same, so V and V do span the same set of vectors, and we can generate any vector of the form x = α + α / 8/. Is the vector r = in span We want to find α and α such that α / /, + α 8/ This leads us to the following system of equations:? What about the vector r =? 8/ α = α = = α = α 8 α = Now we could take the values from the first and third equations and verify that they work in the second and fourth equations as well (and if they don t then r is not in the span). This is fine for lower dimensional vectors, but it isn t the most elegant way of doing it in higher dimensions with more vectors. Instead, we can solve this system the way we have before by forming our augmented matrix and then putting it into rref (note that when we do this we end up with the vectors in the columns instead of the rows as we had before): / 8/ rref. (This isn t quite rref since not all of the rows of zeros are at the bottom of the matrix, but that doesn t matter for what we re doing here.) Now we read off the solution of α =, α =. We repeat this process for the vector r. Putting the vectors into a matrix and row reducing yields rref / 8/ The third row says that =, which is a contradiction, so r is not in the span of these vectors.. (a) Find the span of the vectors v = 5, v =, v =, v = 8 6

7 (b) Is the vector in span{ v, v, v, v }? What about the vector. If possible, write 5 as a linear combination of the vectors,, 5. 7?. Show that. Show that span,, = span,. 6 span,,, 5 = span,,. 6 Linear (In)Dependence Closely related to span and spanning sets is the notion of linear independence. Definition. Vectors v, v,..., v n are said to be linearly dependent if there exist scalars α, α,..., α n not all zero such that α v + α v + + α n v n =. Theorem. The vectors v, v,..., v n (with k ) are linearly dependent if and only if at least one v i is a linear combination of the others. Definition 5. Vectors v, v,..., v n are said to be linearly independent if there does not exist scalars α, α,..., α n not all zero such that α v + α v + + α n v n =. To determine whether a set of vectors is linearly dependent, we will attempt to solve the equation α v + α v + + α n v n = using the same methods we did when trying to determine whether a vector is in the span of a set of vectors: set up the augmented matrix (putting the vectors in as columns) and put it into rref. If we don t get a contradiction in the resulting solutions (saying that = (something nonzero) then they are dependent. If there is a contradiction, then they re independent. Determine whether each set of vectors is linearly dependent or independent. If dependent, then express one of the vectors as a linear combination of the other(s).. { }, 8 7

8 { } / c.,, where c is any real number. 6c 5.,,.,, 7 Determinants (Discussed in sections. and. of FMA.) You ve seen and determinants in Calc III and we ll go over those again in class as well as how to find determinants of higher dimension matrices. Section. in FMA also goes over the general process. With the time remaining in this semester, there are only two properties of determinants on which we ll focus:. Given an n n matrix A, A is invertible if and only if det(a).. If we have a set of n n-dimensional vectors, the the determinant of the matrix formed by those vectors (putting them into the matrix as either columns or rows it doesn t matter which) is if and only if the vectors are linearly dependent (so a nonzero determinant means that they are linearly independent). Note: While both of these are very useful and important from a theoretical perspective, from a computational perspective they are not as useful since computing determinants requires significantly more operations than putting a matrix into rref. Also, the second property is only useful in that specific case if the dimension doesn t match the number of vectors then we need to use the techniques we used when finding spans.. Work through the following exercises from FMA section.:,,,,. 8

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

This MUST hold matrix multiplication satisfies the distributive property.

This MUST hold matrix multiplication satisfies the distributive property. The columns of AB are combinations of the columns of A. The reason is that each column of AB equals A times the corresponding column of B. But that is a linear combination of the columns of A with coefficients

More information

Lecture 2 Systems of Linear Equations and Matrices, Continued

Lecture 2 Systems of Linear Equations and Matrices, Continued Lecture 2 Systems of Linear Equations and Matrices, Continued Math 19620 Outline of Lecture Algorithm for putting a matrix in row reduced echelon form - i.e. Gauss-Jordan Elimination Number of Solutions

More information

Section Gaussian Elimination

Section Gaussian Elimination Section. - Gaussian Elimination A matrix is said to be in row echelon form (REF) if it has the following properties:. The first nonzero entry in any row is a. We call this a leading one or pivot one..

More information

Chapter 4. Solving Systems of Equations. Chapter 4

Chapter 4. Solving Systems of Equations. Chapter 4 Solving Systems of Equations 3 Scenarios for Solutions There are three general situations we may find ourselves in when attempting to solve systems of equations: 1 The system could have one unique solution.

More information

4 Elementary matrices, continued

4 Elementary matrices, continued 4 Elementary matrices, continued We have identified 3 types of row operations and their corresponding elementary matrices. If you check the previous examples, you ll find that these matrices are constructed

More information

Chapter 1. Vectors, Matrices, and Linear Spaces

Chapter 1. Vectors, Matrices, and Linear Spaces 1.4 Solving Systems of Linear Equations 1 Chapter 1. Vectors, Matrices, and Linear Spaces 1.4. Solving Systems of Linear Equations Note. We give an algorithm for solving a system of linear equations (called

More information

LECTURES 4/5: SYSTEMS OF LINEAR EQUATIONS

LECTURES 4/5: SYSTEMS OF LINEAR EQUATIONS LECTURES 4/5: SYSTEMS OF LINEAR EQUATIONS MA1111: LINEAR ALGEBRA I, MICHAELMAS 2016 1 Linear equations We now switch gears to discuss the topic of solving linear equations, and more interestingly, systems

More information

Notes on Row Reduction

Notes on Row Reduction Notes on Row Reduction Francis J. Narcowich Department of Mathematics Texas A&M University September The Row-Reduction Algorithm The row-reduced form of a matrix contains a great deal of information, both

More information

Math Computation Test 1 September 26 th, 2016 Debate: Computation vs. Theory Whatever wins, it ll be Huuuge!

Math Computation Test 1 September 26 th, 2016 Debate: Computation vs. Theory Whatever wins, it ll be Huuuge! Math 5- Computation Test September 6 th, 6 Debate: Computation vs. Theory Whatever wins, it ll be Huuuge! Name: Answer Key: Making Math Great Again Be sure to show your work!. (8 points) Consider the following

More information

MATH 54 - WORKSHEET 1 MONDAY 6/22

MATH 54 - WORKSHEET 1 MONDAY 6/22 MATH 54 - WORKSHEET 1 MONDAY 6/22 Row Operations: (1 (Replacement Add a multiple of one row to another row. (2 (Interchange Swap two rows. (3 (Scaling Multiply an entire row by a nonzero constant. A matrix

More information

Linear Equations in Linear Algebra

Linear Equations in Linear Algebra 1 Linear Equations in Linear Algebra 1.1 SYSTEMS OF LINEAR EQUATIONS LINEAR EQUATION x 1,, x n A linear equation in the variables equation that can be written in the form a 1 x 1 + a 2 x 2 + + a n x n

More information

4 Elementary matrices, continued

4 Elementary matrices, continued 4 Elementary matrices, continued We have identified 3 types of row operations and their corresponding elementary matrices. To repeat the recipe: These matrices are constructed by performing the given row

More information

Lecture 4: Gaussian Elimination and Homogeneous Equations

Lecture 4: Gaussian Elimination and Homogeneous Equations Lecture 4: Gaussian Elimination and Homogeneous Equations Reduced Row Echelon Form An augmented matrix associated to a system of linear equations is said to be in Reduced Row Echelon Form (RREF) if the

More information

Review for Exam Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions.

Review for Exam Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions. Review for Exam. Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions. x + y z = 2 x + 2y + z = 3 x + y + (a 2 5)z = a 2 The augmented matrix for

More information

1 Last time: inverses

1 Last time: inverses MATH Linear algebra (Fall 8) Lecture 8 Last time: inverses The following all mean the same thing for a function f : X Y : f is invertible f is one-to-one and onto 3 For each b Y there is exactly one a

More information

Lecture 3: Gaussian Elimination, continued. Lecture 3: Gaussian Elimination, continued

Lecture 3: Gaussian Elimination, continued. Lecture 3: Gaussian Elimination, continued Definition The process of solving a system of linear equations by converting the system to an augmented matrix is called Gaussian Elimination. The general strategy is as follows: Convert the system of

More information

Lecture 6 & 7. Shuanglin Shao. September 16th and 18th, 2013

Lecture 6 & 7. Shuanglin Shao. September 16th and 18th, 2013 Lecture 6 & 7 Shuanglin Shao September 16th and 18th, 2013 1 Elementary matrices 2 Equivalence Theorem 3 A method of inverting matrices Def An n n matrice is called an elementary matrix if it can be obtained

More information

Example: 2x y + 3z = 1 5y 6z = 0 x + 4z = 7. Definition: Elementary Row Operations. Example: Type I swap rows 1 and 3

Example: 2x y + 3z = 1 5y 6z = 0 x + 4z = 7. Definition: Elementary Row Operations. Example: Type I swap rows 1 and 3 Math 0 Row Reduced Echelon Form Techniques for solving systems of linear equations lie at the heart of linear algebra. In high school we learn to solve systems with or variables using elimination and substitution

More information

MATH240: Linear Algebra Review for exam #1 6/10/2015 Page 1

MATH240: Linear Algebra Review for exam #1 6/10/2015 Page 1 MATH24: Linear Algebra Review for exam # 6//25 Page No review sheet can cover everything that is potentially fair game for an exam, but I tried to hit on all of the topics with these questions, as well

More information

Methods for Solving Linear Systems Part 2

Methods for Solving Linear Systems Part 2 Methods for Solving Linear Systems Part 2 We have studied the properties of matrices and found out that there are more ways that we can solve Linear Systems. In Section 7.3, we learned that we can use

More information

Elementary matrices, continued. To summarize, we have identified 3 types of row operations and their corresponding

Elementary matrices, continued. To summarize, we have identified 3 types of row operations and their corresponding Elementary matrices, continued To summarize, we have identified 3 types of row operations and their corresponding elementary matrices. If you check the previous examples, you ll find that these matrices

More information

Lecture 12: Solving Systems of Linear Equations by Gaussian Elimination

Lecture 12: Solving Systems of Linear Equations by Gaussian Elimination Lecture 12: Solving Systems of Linear Equations by Gaussian Elimination Winfried Just, Ohio University September 22, 2017 Review: The coefficient matrix Consider a system of m linear equations in n variables.

More information

3.4 Elementary Matrices and Matrix Inverse

3.4 Elementary Matrices and Matrix Inverse Math 220: Summer 2015 3.4 Elementary Matrices and Matrix Inverse A n n elementary matrix is a matrix which is obtained from the n n identity matrix I n n by a single elementary row operation. Elementary

More information

Midterm 1 Review. Written by Victoria Kala SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015

Midterm 1 Review. Written by Victoria Kala SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015 Midterm 1 Review Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015 Summary This Midterm Review contains notes on sections 1.1 1.5 and 1.7 in your

More information

Linear Algebra Practice Problems

Linear Algebra Practice Problems Math 7, Professor Ramras Linear Algebra Practice Problems () Consider the following system of linear equations in the variables x, y, and z, in which the constants a and b are real numbers. x y + z = a

More information

Row Reduced Echelon Form

Row Reduced Echelon Form Math 40 Row Reduced Echelon Form Solving systems of linear equations lies at the heart of linear algebra. In high school we learn to solve systems in or variables using elimination and substitution of

More information

The definition of a vector space (V, +, )

The definition of a vector space (V, +, ) The definition of a vector space (V, +, ) 1. For any u and v in V, u + v is also in V. 2. For any u and v in V, u + v = v + u. 3. For any u, v, w in V, u + ( v + w) = ( u + v) + w. 4. There is an element

More information

Lecture 9: Elementary Matrices

Lecture 9: Elementary Matrices Lecture 9: Elementary Matrices Review of Row Reduced Echelon Form Consider the matrix A and the vector b defined as follows: 1 2 1 A b 3 8 5 A common technique to solve linear equations of the form Ax

More information

EXAM 2 REVIEW DAVID SEAL

EXAM 2 REVIEW DAVID SEAL EXAM 2 REVIEW DAVID SEAL 3. Linear Systems and Matrices 3.2. Matrices and Gaussian Elimination. At this point in the course, you all have had plenty of practice with Gaussian Elimination. Be able to row

More information

Matrices and RRE Form

Matrices and RRE Form Matrices and RRE Form Notation R is the real numbers, C is the complex numbers (we will only consider complex numbers towards the end of the course) is read as an element of For instance, x R means that

More information

Example: 2x y + 3z = 1 5y 6z = 0 x + 4z = 7. Definition: Elementary Row Operations. Example: Type I swap rows 1 and 3

Example: 2x y + 3z = 1 5y 6z = 0 x + 4z = 7. Definition: Elementary Row Operations. Example: Type I swap rows 1 and 3 Linear Algebra Row Reduced Echelon Form Techniques for solving systems of linear equations lie at the heart of linear algebra. In high school we learn to solve systems with or variables using elimination

More information

GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511)

GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511) GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511) D. ARAPURA Gaussian elimination is the go to method for all basic linear classes including this one. We go summarize the main ideas. 1.

More information

Last Time. x + 3y = 6 x + 2y = 1. x + 3y = 6 y = 1. 2x + 4y = 8 x 2y = 1. x + 3y = 6 2x y = 7. Lecture 2

Last Time. x + 3y = 6 x + 2y = 1. x + 3y = 6 y = 1. 2x + 4y = 8 x 2y = 1. x + 3y = 6 2x y = 7. Lecture 2 January 9 Last Time 1. Last time we ended with saying that the following four systems are equivalent in the sense that we can move from one system to the other by a special move we discussed. (a) (b) (c)

More information

Elementary Matrices. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics

Elementary Matrices. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics Elementary Matrices MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Outline Today s discussion will focus on: elementary matrices and their properties, using elementary

More information

Lecture 22: Section 4.7

Lecture 22: Section 4.7 Lecture 22: Section 47 Shuanglin Shao December 2, 213 Row Space, Column Space, and Null Space Definition For an m n, a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn, the vectors r 1 = [ a 11 a 12 a 1n

More information

Linear Independence Reading: Lay 1.7

Linear Independence Reading: Lay 1.7 Linear Independence Reading: Lay 17 September 11, 213 In this section, we discuss the concept of linear dependence and independence I am going to introduce the definitions and then work some examples and

More information

Chapter 1: Linear Equations

Chapter 1: Linear Equations Chapter : Linear Equations (Last Updated: September, 6) The material for these notes is derived primarily from Linear Algebra and its applications by David Lay (4ed).. Systems of Linear Equations Before

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

Math 3C Lecture 20. John Douglas Moore

Math 3C Lecture 20. John Douglas Moore Math 3C Lecture 20 John Douglas Moore May 18, 2009 TENTATIVE FORMULA I Midterm I: 20% Midterm II: 20% Homework: 10% Quizzes: 10% Final: 40% TENTATIVE FORMULA II Higher of two midterms: 30% Homework: 10%

More information

Math 54 HW 4 solutions

Math 54 HW 4 solutions Math 54 HW 4 solutions 2.2. Section 2.2 (a) False: Recall that performing a series of elementary row operations A is equivalent to multiplying A by a series of elementary matrices. Suppose that E,...,

More information

Kevin James. MTHSC 3110 Section 2.2 Inverses of Matrices

Kevin James. MTHSC 3110 Section 2.2 Inverses of Matrices MTHSC 3110 Section 2.2 Inverses of Matrices Definition Suppose that T : R n R m is linear. We will say that T is invertible if for every b R m there is exactly one x R n so that T ( x) = b. Note If T is

More information

Linear Equation: a 1 x 1 + a 2 x a n x n = b. x 1, x 2,..., x n : variables or unknowns

Linear Equation: a 1 x 1 + a 2 x a n x n = b. x 1, x 2,..., x n : variables or unknowns Linear Equation: a x + a 2 x 2 +... + a n x n = b. x, x 2,..., x n : variables or unknowns a, a 2,..., a n : coefficients b: constant term Examples: x + 4 2 y + (2 5)z = is linear. x 2 + y + yz = 2 is

More information

Chapter 1: Linear Equations

Chapter 1: Linear Equations Chapter : Linear Equations (Last Updated: September, 7) The material for these notes is derived primarily from Linear Algebra and its applications by David Lay (4ed).. Systems of Linear Equations Before

More information

Determine whether the following system has a trivial solution or non-trivial solution:

Determine whether the following system has a trivial solution or non-trivial solution: Practice Questions Lecture # 7 and 8 Question # Determine whether the following system has a trivial solution or non-trivial solution: x x + x x x x x The coefficient matrix is / R, R R R+ R The corresponding

More information

Elementary Linear Algebra

Elementary Linear Algebra Elementary Linear Algebra Linear algebra is the study of; linear sets of equations and their transformation properties. Linear algebra allows the analysis of; rotations in space, least squares fitting,

More information

Solving Linear Systems Using Gaussian Elimination

Solving Linear Systems Using Gaussian Elimination Solving Linear Systems Using Gaussian Elimination DEFINITION: A linear equation in the variables x 1,..., x n is an equation that can be written in the form a 1 x 1 +...+a n x n = b, where a 1,...,a n

More information

Row Reduction and Echelon Forms

Row Reduction and Echelon Forms Row Reduction and Echelon Forms 1 / 29 Key Concepts row echelon form, reduced row echelon form pivot position, pivot, pivot column basic variable, free variable general solution, parametric solution existence

More information

System of Linear Equations

System of Linear Equations Math 20F Linear Algebra Lecture 2 1 System of Linear Equations Slide 1 Definition 1 Fix a set of numbers a ij, b i, where i = 1,, m and j = 1,, n A system of m linear equations in n variables x j, is given

More information

Linear Equations in Linear Algebra

Linear Equations in Linear Algebra 1 Linear Equations in Linear Algebra 1.1 SYSTEMS OF LINEAR EQUATIONS LINEAR EQUATION,, 1 n A linear equation in the variables equation that can be written in the form a a a b 1 1 2 2 n n a a is an where

More information

MATH 310, REVIEW SHEET 2

MATH 310, REVIEW SHEET 2 MATH 310, REVIEW SHEET 2 These notes are a very short summary of the key topics in the book (and follow the book pretty closely). You should be familiar with everything on here, but it s not comprehensive,

More information

MATH 221: SOLUTIONS TO SELECTED HOMEWORK PROBLEMS

MATH 221: SOLUTIONS TO SELECTED HOMEWORK PROBLEMS MATH 221: SOLUTIONS TO SELECTED HOMEWORK PROBLEMS 1. HW 1: Due September 4 1.1.21. Suppose v, w R n and c is a scalar. Prove that Span(v + cw, w) = Span(v, w). We must prove two things: that every element

More information

MAC1105-College Algebra. Chapter 5-Systems of Equations & Matrices

MAC1105-College Algebra. Chapter 5-Systems of Equations & Matrices MAC05-College Algebra Chapter 5-Systems of Equations & Matrices 5. Systems of Equations in Two Variables Solving Systems of Two Linear Equations/ Two-Variable Linear Equations A system of equations is

More information

Review Solutions for Exam 1

Review Solutions for Exam 1 Definitions Basic Theorems. Finish the definition: Review Solutions for Exam (a) A linear combination of vectors {v,..., v n } is: any vector of the form c v + c v + + c n v n (b) A set of vectors {v,...,

More information

Pre-Calculus I. For example, the system. x y 2 z. may be represented by the augmented matrix

Pre-Calculus I. For example, the system. x y 2 z. may be represented by the augmented matrix Pre-Calculus I 8.1 Matrix Solutions to Linear Systems A matrix is a rectangular array of elements. o An array is a systematic arrangement of numbers or symbols in rows and columns. Matrices (the plural

More information

Section 1.1 System of Linear Equations. Dr. Abdulla Eid. College of Science. MATHS 211: Linear Algebra

Section 1.1 System of Linear Equations. Dr. Abdulla Eid. College of Science. MATHS 211: Linear Algebra Section 1.1 System of Linear Equations College of Science MATHS 211: Linear Algebra (University of Bahrain) Linear System 1 / 33 Goals:. 1 Define system of linear equations and their solutions. 2 To represent

More information

INVERSE OF A MATRIX [2.2]

INVERSE OF A MATRIX [2.2] INVERSE OF A MATRIX [2.2] The inverse of a matrix: Introduction We have a mapping from R n to R n represented by a matrix A. Can we invert this mapping? i.e. can we find a matrix (call it B for now) such

More information

Math 4377/6308 Advanced Linear Algebra

Math 4377/6308 Advanced Linear Algebra 3.1 Elementary Matrix Math 4377/6308 Advanced Linear Algebra 3.1 Elementary Matrix Operations and Elementary Matrix Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/

More information

MTH Linear Algebra. Study Guide. Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education

MTH Linear Algebra. Study Guide. Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education MTH 3 Linear Algebra Study Guide Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education June 3, ii Contents Table of Contents iii Matrix Algebra. Real Life

More information

1300 Linear Algebra and Vector Geometry Week 2: Jan , Gauss-Jordan, homogeneous matrices, intro matrix arithmetic

1300 Linear Algebra and Vector Geometry Week 2: Jan , Gauss-Jordan, homogeneous matrices, intro matrix arithmetic 1300 Linear Algebra and Vector Geometry Week 2: Jan 14 18 1.2, 1.3... Gauss-Jordan, homogeneous matrices, intro matrix arithmetic R. Craigen Office: MH 523 Email: craigenr@umanitoba.ca Winter 2019 What

More information

Final Exam Practice Problems Answers Math 24 Winter 2012

Final Exam Practice Problems Answers Math 24 Winter 2012 Final Exam Practice Problems Answers Math 4 Winter 0 () The Jordan product of two n n matrices is defined as A B = (AB + BA), where the products inside the parentheses are standard matrix product. Is the

More information

MTH 2032 Semester II

MTH 2032 Semester II MTH 232 Semester II 2-2 Linear Algebra Reference Notes Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education December 28, 2 ii Contents Table of Contents

More information

Solutions of Linear system, vector and matrix equation

Solutions of Linear system, vector and matrix equation Goals: Solutions of Linear system, vector and matrix equation Solutions of linear system. Vectors, vector equation. Matrix equation. Math 112, Week 2 Suggested Textbook Readings: Sections 1.3, 1.4, 1.5

More information

Math 314H EXAM I. 1. (28 points) The row reduced echelon form of the augmented matrix for the system. is the matrix

Math 314H EXAM I. 1. (28 points) The row reduced echelon form of the augmented matrix for the system. is the matrix Math 34H EXAM I Do all of the problems below. Point values for each of the problems are adjacent to the problem number. Calculators may be used to check your answer but not to arrive at your answer. That

More information

Exam 2 Solutions. (a) Is W closed under addition? Why or why not? W is not closed under addition. For example,

Exam 2 Solutions. (a) Is W closed under addition? Why or why not? W is not closed under addition. For example, Exam 2 Solutions. Let V be the set of pairs of real numbers (x, y). Define the following operations on V : (x, y) (x, y ) = (x + x, xx + yy ) r (x, y) = (rx, y) Check if V together with and satisfy properties

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

2 Systems of Linear Equations

2 Systems of Linear Equations 2 Systems of Linear Equations A system of equations of the form or is called a system of linear equations. x + 2y = 7 2x y = 4 5p 6q + r = 4 2p + 3q 5r = 7 6p q + 4r = 2 Definition. An equation involving

More information

Lecture 1 Systems of Linear Equations and Matrices

Lecture 1 Systems of Linear Equations and Matrices Lecture 1 Systems of Linear Equations and Matrices Math 19620 Outline of Course Linear Equations and Matrices Linear Transformations, Inverses Bases, Linear Independence, Subspaces Abstract Vector Spaces

More information

36 What is Linear Algebra?

36 What is Linear Algebra? 36 What is Linear Algebra? The authors of this textbook think that solving linear systems of equations is a big motivation for studying linear algebra This is certainly a very respectable opinion as systems

More information

5x 2 = 10. x 1 + 7(2) = 4. x 1 3x 2 = 4. 3x 1 + 9x 2 = 8

5x 2 = 10. x 1 + 7(2) = 4. x 1 3x 2 = 4. 3x 1 + 9x 2 = 8 1 To solve the system x 1 + x 2 = 4 2x 1 9x 2 = 2 we find an (easier to solve) equivalent system as follows: Replace equation 2 with (2 times equation 1 + equation 2): x 1 + x 2 = 4 Solve equation 2 for

More information

Homework 1 Due: Wednesday, August 27. x + y + z = 1. x y = 3 x + y + z = c 2 2x + cz = 4

Homework 1 Due: Wednesday, August 27. x + y + z = 1. x y = 3 x + y + z = c 2 2x + cz = 4 Homework 1 Due: Wednesday, August 27 1. Find all values of c for which the linear system: (a) has no solutions. (b) has exactly one solution. (c) has infinitely many solutions. (d) is consistent. x + y

More information

Math 4A Notes. Written by Victoria Kala Last updated June 11, 2017

Math 4A Notes. Written by Victoria Kala Last updated June 11, 2017 Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...

More information

Linear Algebra: Sample Questions for Exam 2

Linear Algebra: Sample Questions for Exam 2 Linear Algebra: Sample Questions for Exam 2 Instructions: This is not a comprehensive review: there are concepts you need to know that are not included. Be sure you study all the sections of the book and

More information

Section 6.3. Matrices and Systems of Equations

Section 6.3. Matrices and Systems of Equations Section 6.3 Matrices and Systems of Equations Introduction Definitions A matrix is a rectangular array of numbers. Definitions A matrix is a rectangular array of numbers. For example: [ 4 7 π 3 2 5 Definitions

More information

1 - Systems of Linear Equations

1 - Systems of Linear Equations 1 - Systems of Linear Equations 1.1 Introduction to Systems of Linear Equations Almost every problem in linear algebra will involve solving a system of equations. ü LINEAR EQUATIONS IN n VARIABLES We are

More information

Designing Information Devices and Systems I Fall 2018 Lecture Notes Note 6

Designing Information Devices and Systems I Fall 2018 Lecture Notes Note 6 EECS 16A Designing Information Devices and Systems I Fall 2018 Lecture Notes Note 6 6.1 Introduction: Matrix Inversion In the last note, we considered a system of pumps and reservoirs where the water in

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

is a 3 4 matrix. It has 3 rows and 4 columns. The first row is the horizontal row [ ]

is a 3 4 matrix. It has 3 rows and 4 columns. The first row is the horizontal row [ ] Matrices: Definition: An m n matrix, A m n is a rectangular array of numbers with m rows and n columns: a, a, a,n a, a, a,n A m,n =...... a m, a m, a m,n Each a i,j is the entry at the i th row, j th column.

More information

MAC Module 3 Determinants. Learning Objectives. Upon completing this module, you should be able to:

MAC Module 3 Determinants. Learning Objectives. Upon completing this module, you should be able to: MAC 2 Module Determinants Learning Objectives Upon completing this module, you should be able to:. Determine the minor, cofactor, and adjoint of a matrix. 2. Evaluate the determinant of a matrix by cofactor

More information

22A-2 SUMMER 2014 LECTURE 5

22A-2 SUMMER 2014 LECTURE 5 A- SUMMER 0 LECTURE 5 NATHANIEL GALLUP Agenda Elimination to the identity matrix Inverse matrices LU factorization Elimination to the identity matrix Previously, we have used elimination to get a system

More information

Linear Algebra for Beginners Open Doors to Great Careers. Richard Han

Linear Algebra for Beginners Open Doors to Great Careers. Richard Han Linear Algebra for Beginners Open Doors to Great Careers Richard Han Copyright 2018 Richard Han All rights reserved. CONTENTS PREFACE... 7 1 - INTRODUCTION... 8 2 SOLVING SYSTEMS OF LINEAR EQUATIONS...

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

Chapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations

Chapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2

More information

Lectures on Linear Algebra for IT

Lectures on Linear Algebra for IT Lectures on Linear Algebra for IT by Mgr. Tereza Kovářová, Ph.D. following content of lectures by Ing. Petr Beremlijski, Ph.D. Department of Applied Mathematics, VSB - TU Ostrava Czech Republic 2. Systems

More information

Math 4377/6308 Advanced Linear Algebra

Math 4377/6308 Advanced Linear Algebra 1.4 Linear Combinations Math 4377/6308 Advanced Linear Algebra 1.4 Linear Combinations & Systems of Linear Equations Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/

More information

9.1 - Systems of Linear Equations: Two Variables

9.1 - Systems of Linear Equations: Two Variables 9.1 - Systems of Linear Equations: Two Variables Recall that a system of equations consists of two or more equations each with two or more variables. A solution to a system in two variables is an ordered

More information

Section 2.2: The Inverse of a Matrix

Section 2.2: The Inverse of a Matrix Section 22: The Inverse of a Matrix Recall that a linear equation ax b, where a and b are scalars and a 0, has the unique solution x a 1 b, where a 1 is the reciprocal of a From this result, it is natural

More information

Introduction to Determinants

Introduction to Determinants Introduction to Determinants For any square matrix of order 2, we have found a necessary and sufficient condition for invertibility. Indeed, consider the matrix The matrix A is invertible if and only if.

More information

Systems of Linear Equations. By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University

Systems of Linear Equations. By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University Systems of Linear Equations By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University Standard of Competency: Understanding the properties of systems of linear equations, matrices,

More information

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

Chapter 5. Linear Algebra. Sections A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form Chapter 5. Linear Algebra Sections 5.1 5.3 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

More information

a s 1.3 Matrix Multiplication. Know how to multiply two matrices and be able to write down the formula

a s 1.3 Matrix Multiplication. Know how to multiply two matrices and be able to write down the formula Syllabus for Math 308, Paul Smith Book: Kolman-Hill Chapter 1. Linear Equations and Matrices 1.1 Systems of Linear Equations Definition of a linear equation and a solution to a linear equations. Meaning

More information

Math 344 Lecture # Linear Systems

Math 344 Lecture # Linear Systems Math 344 Lecture #12 2.7 Linear Systems Through a choice of bases S and T for finite dimensional vector spaces V (with dimension n) and W (with dimension m), a linear equation L(v) = w becomes the linear

More information

Mon Feb Matrix inverses, the elementary matrix approach overview of skipped section 2.5. Announcements: Warm-up Exercise:

Mon Feb Matrix inverses, the elementary matrix approach overview of skipped section 2.5. Announcements: Warm-up Exercise: Math 2270-004 Week 6 notes We will not necessarily finish the material from a given day's notes on that day We may also add or subtract some material as the week progresses, but these notes represent an

More information

REPLACE ONE ROW BY ADDING THE SCALAR MULTIPLE OF ANOTHER ROW

REPLACE ONE ROW BY ADDING THE SCALAR MULTIPLE OF ANOTHER ROW 20 CHAPTER 1 Systems of Linear Equations REPLACE ONE ROW BY ADDING THE SCALAR MULTIPLE OF ANOTHER ROW The last type of operation is slightly more complicated. Suppose that we want to write down the elementary

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

Working with Block Structured Matrices

Working with Block Structured Matrices Working with Block Structured Matrices Numerical linear algebra lies at the heart of modern scientific computing and computational science. Today it is not uncommon to perform numerical computations with

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

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

Row Reduction

Row Reduction Row Reduction 1-12-2015 Row reduction (or Gaussian elimination) is the process of using row operations to reduce a matrix to row reduced echelon form This procedure is used to solve systems of linear equations,

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 Introduction: linear equations Read 1.1 (in the text that is!) Go to course, class webpages.

More information