Roberto s Notes on Linear Algebra Chapter 3: Linear systems and matrices Section 4 Using matrices to represent linear systems What you need to know already: What a linear system is. What elementary operations on a system are and why they are useful. What you can learn here: A more efficient way to represent a system that leads to a faster and easier way to solve it. One motivation behind the very important mathematical concept of matrix. At the end of the last section I pointed out that when performing elementary operations, we are really only using the coefficients of the equations involved. I also suggested that the original system notation hides this fact and hinders the efficiency of the whole method. Therefore, we may want to use a different notation that focuses on the coefficients and hides the variables, while keeping track of them. the coefficients, organized so as to clearly indicate their corresponding variables, and the constants. Strategy for better representing a linear system A linear system of the form: a11x1 a12 x2... a1 nxn c1 a21x1 a22x2... a2nxn c2... a 1x1 a 2x2... a x c m m mn n m can be represented more efficiently by eliminating the variables and the equal signs and keeping only But if we throw away all these pieces, don t we end up with a confusing pile of numbers? Yes, but the pile can be given order and meaning with a few small agreements. We start with a simple, but tremendously powerful definition. Definition A matrix is an ordered set of m nscalars, called its entries, organized in m rows and n columns. The standard notation for a matrix and its rows and columns is: Linear Algebra Chapter 3: Linear systems and matrices Section 4: Using matrices to represent linear systems Page 1
a11 a12... a1 n a21 a22... a 2n A...... aij... am1 am2... amn a1 j a 2 j Ri ai 1 ai 2... ain C j... amj So a matrix is like a vector, only organized in 2 dimensions! Exactly! In later sections and chapters we shall discuss matrices and their properties in great detail and you will see why I say that they are a very powerful mathematical tool. What you see in this definition is all we need to complete our work on solving systems in an efficient way. Oh, I can see that if we just focus on the coefficients and constants of a system we are left with a matrix! Definition Given a linear system of the form: a x a x... a x c a x a x... a x c... a x a x... a x c 11 1 12 2 1n n 1 21 1 22 2 2n n 2 m1 1 m2 2 mn n m its matrix of coefficients is: a11 a12... a1 n a21 a22... a 2n............ am1 am2... amn its augmented matrix is: a11 a12... a1 n c1 a21 a22... a2n c 2............... am1 am2... amn cm aug what? The word augmented comes from Latin and it means increased or made bigger. In this case we start with the matrix of coefficients and make it bigger by inserting the column of constants at the end. This augmentation is a little trick that we shall use again in other situations. In fact, there is another piece of notation that is often used in conjunction with augmented matrices: Linear Algebra Chapter 3: Linear systems and matrices Section 4: Using matrices to represent linear systems Page 2
Knot on your finger In order to emphasize that a given matrix is the augmented matrix of a system, it is good practice but not essential to draw a vertical line between the coefficients and the constants: a11 a12... a1 n c1 a21 a22... a2n c2 A c............... am1 am2... amn cm Example: 3x 5y 8z 3 5x y 2z 2 8x 4y 6z 5 In this system, the vector of variables is 3 5 constants is c 2. x x y z, while the vector of The matrix of coefficients and the augmented matrix are therefore, respectively: Example: 3 5 8 3 5 8 3 A 5 1 2, A c 5 1 2 2 8 4 6 8 4 6 5 2 1 3 2 5 2 6 2 1 0 4 3 If this is the augmented matrix of a linear system, then the system is 2x y 3z 2 5x 2y 6z 2 x 4z 3 Notice that there is a clear and recognizable correspondence between the rows and columns of the matrices of a system and the key features of the system. Knots on your finger Each equation of a linear system corresponds to one row of its augmented matrix. Each variable of a linear system is represented through one column of its matrix of coefficients or its augmented matrix. Linear Algebra Chapter 3: Linear systems and matrices Section 4: Using matrices to represent linear systems Page 3
Example: 2x y 3z 2 5x 2y 6z 2 x 4z3 If we look at this system and its augmented matrix: we can see that: 2 1 3 2 5 2 6 2 1 0 4 3 the first equation is represented by the first row of the matrix, the second row of the matrix described the second equation of the system, the coefficients of the variable y are all given by the second column, the third column consists of the coefficients of the variable z. and so on. I can see how this notation saves some ink or lead, but how it is more efficient? This whole issue of efficiency came up in relation to elementary operations and these operations become truly a great tool when we apply them to the matrix of a system. Definition An elementary row operation (ERO) is a change we make to a matrix by following one of these three rules: Switch the position of two rows: Rh R k. Multiply one row by a non-zero scalar: cr h. Add to a row a multiple of another row: Rh cr k. Aren t these the same as elementary operations on a system? Definitely, but applied to matrices. And what a useful extension this is! But before I show you its full power, here is how the solution of the system used in Example 3.3.6 is obtained by using ERO s instead of keeping the system: Example: 2x 3y z 25 x 2y 4z 25 3x y 2z 2 Starting from this same original system, we write its augmented matrix and perform the ERO s corresponding to the elementary operations we used before: 2 3 1 25 R1 R21 2 4 25 1 2 4 25 2 3 1 25 3 1 2 2 3 1 2 2 R R1 R2 R1 1 1 2 4 25 2 1 2 4 25 2 3 1 25 0 1 9 25 3 1 2 2 3 1 2 2 3R 1 2 4 25 1 1 2 4 25 0 1 9 25 7 0 1 9 25 0 7 14 77 0 1 2 11 3 1 R3 Linear Algebra Chapter 3: Linear systems and matrices Section 4: Using matrices to represent linear systems Page 4
R3 R21 2 4 25 1 1 2 4 25 R3 0 1 9 25 7 0 1 9 25 0 0 7 14 0 0 1 2 R2 9R3 1 2 4 25 1 R2 1 2 4 25 0 1 0 7 0 1 0 7 0 0 1 2 0 0 1 2 And finally: R1 4R3 1 2 0 17 R1 2R2 1 0 0 3 0 1 0 7 0 1 0 7 0 0 1 2 0 0 1 2 How does this last augmented matrix read, as a system? which is our solution! x 3 y 7 z 2 That seems obvious! Definition Two matrices are said to be row equivalent if one can be obtained from the other through a sequence of ERO s. Technical fact Two matrices are row equivalent if and only if the systems for which they are augmented matrices are equivalent. Good, since I am including the proof of this fact in the Learning questions. It is easy, but make sure you use proper jargon, notation and logic! I want to finish this section with a few reflections on homogeneous systems. Notice that: each linear system has an augmented matrix; any matrix can be considered as the augmented matrix of a system; each elementary operation performed on a system corresponds to the respective ERO done on the augmented matrix. Because of this strict correspondence we shall use the following terminology: Knots on your finger The last column of the augmented matrix of a homogeneous system consists only of 0 s. For this reason, such column is usually omitted, so that the augmented matrix of a homogeneous system is written in the same way as its matrix of coefficients However, do not let the convenient notation confuse you: remember that the augmented matrix Linear Algebra Chapter 3: Linear systems and matrices Section 4: Using matrices to represent linear systems Page 5
and the matrix of coefficients are different entities with different meanings. Maybe I need some practice, but I am still not convinced that this new notation is that useful. Right on both counts! You do need some practice, and the Checkpoint is coming up. But it is also true that the full power of the matrix notation is still not visible. That is because we have not looked yet at how to use ERO s in a systematic way to solve a system and have not stopped to reflect on what it is that we are looking for, from the matrix, in order to obtain such solution in an effective way. That will be addressed in the next section. Summary A matrix is basically a two-dimensional version on the concept of vector: a set of numbers ordered in rows and columns. Any linear system can be represented by a matrix, where the columns represent variables and constants, while the rows represent equations. Elementary operations are immediately translated into elementary row operations (ERO) and will soon be used to solve systems in an easy way. By the way: matrices are an incredibly useful and used tool in most areas of mathematics. Common errors to avoid We are starting to encounter more and more new jargon words and expressions: work on them, learn their meaning and remember that in linear algebra the concepts they denote are more crucial and complex than the computational methods. Do not underestimate the need to focus on them, or you will be sorry you did! Learning questions for Section LA 3-4 Review questions: 1. Describe what a matrix is. 2. Identify the relationships between a system and the rows and columns of its representing matrix. 3. Explain why the matrix method of representing a system is a great idea. 4. Clarify the difference between the augmented matrix of a system and its matrix of coefficients. 5. Explain the connection between elementary operations and elementary row operations. Linear Algebra Chapter 3: Linear systems and matrices Section 4: Using matrices to represent linear systems Page 6
Memory questions: 1. Which part of the augmented matrix of a linear system corresponds to a variable? 2. Which part of the augmented matrix of a linear system corresponds to an equation? 3. Which part of the augmented matrix of a linear system corresponds to the constants? 4. List the three types of elementary row operations. Computation questions: For each of the systems presented in questions 1-8: a) construct the corresponding matrix of coefficients b) construct the corresponding augmented matrix c) apply one ERO of each of the three types, consecutively, as if attempting to find the solution of the system. 1. 2. 3. 3x 5y 8z 3 5x y 2z 2 8x 4y 6z 5 3x y z 4 x 2y 3z 2 4x y 2z 6 2x 3y z 25 x 2y 4z 25 3x y 2z 2 4. 5. 6. 2x 2z 1 3x y 4z 7 6x y z 0 3x1 2x2 1 4x15 x2 3 7x13x2 0 2x 2z 1 3x y 4z 7 x y 5 7. 8. x y 2z w 1 2x y 2z 2w 2 x y 4z w 2 4x 6z 4w 3 7x1 x2 3x3 2x4 2 x1 x2 2x3 4x4 1 3x1 x2 12x3 16x4 5 Linear Algebra Chapter 3: Linear systems and matrices Section 4: Using matrices to represent linear systems Page 7
For each of the matrices presented in questions 9-11: a) construct the system for which that is the augmented matrix b) construct the homogeneous system for which this is the matrix of coefficient 9. 5 3 1 1 1 0 3 2 4 2 0 7 10. 5 3 1 1 1 0 3 2 4 2 0 7 2 1 3 5 1 11. 0 0 3 1 2 12. Use suitable substitutions to change this system x 3 e y z 7 x 3 6e y 2 z 2 into a linear system, and present the augmented matrix corresponding to such new linear system. Theory questions: 1. Given the matrix 5 3 1 1 1 0 3 2 4 2 0 7, explain why the easiest elementary row operation that generates a 0 in lower left corner is a linear combination of the last two rows. 2. Why do we use only elementary row operations to solve a system? Templated questions: 1. Construct a linear system involving no more than 4 equations and no more than 4 variables and extract from it the matrix of coefficients and the augmented matrix. 2. Construct a matrix consisting of no more than 4 rows and no more than 4 columns and extract from it the linear system for which it is the matrix of coefficients and the one for which it is the augmented matrix. What questions do you have for your instructor? Linear Algebra Chapter 3: Linear systems and matrices Section 4: Using matrices to represent linear systems Page 8