Math 4. Math for Economists. Lecture 02 

This lecture's topic is "Methodology for Solving Linear Systems". Here we are learning about solving Linear Systems with matrices. Hope that helps someone since the labelling of these lectures isn't very detailed.

00:54 Methodology for solving linear systems - using an example 4:30 Organise the coefficients in a matrix 6:16 Solution matrix 8:22 If the objective was to get remove the 2x from the second equation then: the top equation could be multiplied by -2, and then add the first and second equations. This can be applied to the matrix as well. 9:06 Objective is now trying to obtain the first column of the solution matrix (so have a 1 in the top left hand corner, and then 0s in the left middle and left bottom positions of the matrix.) The 1 in the top left hand corner position is called a 'pivot' 11:09 Row 2 (R2) can be replaced by -2*Row 1(R1) + R2 (i.e. R2 --> -2R1+R2) 11:45 Row 3 (R3) can be replaced by R1+R3 12:50 Re-writing the matrix with the new R2 and R3 14:36 Now trying to obtain the 2nd column of the solution matrix (0s in the middle top and middle bottom positions, and 1 in the centre position of the matrix) 15:00 R2 --> -1/3 * R2 (replace row 2 with row 2 divided by -3) 15:43 The 1 in the centre of the matrix will now be used to eliminate the numbers which occur directly above and below it 16:40 R1 --> -R2+R1 (replace row 1 with -1 multiplied row 2 + row 1, or in other words, replace row 1 with row 1 minus row 2) 18:00 R3 --> -3R2 + R3 (replace row 3 with -3 multiplied by row 2 + row 3). We use R2 instead of R1 because otherwise, you will add a number to the 0 that you already obtained from earlier calculations 19:30 All that is left is to deal with the 3rd column. Want to have 0s in the top right and middle right of the matrix. 19:45 R3 --> 1/5R3 (replace row 3 with row 3 divided by 5) 20:14 R2 --> R3 + R2 (replace row 2 with row 3 + row 2), which will put 0 in the middle right position of the matrix 21:05 Arrived at solution matrix 22:00 (Reduced) Row Echelon Form of a Matrix There is a diagonal staggered effect of 1s, beginning in the top left position 25:35 Key features of Row Echelon Form 1) Rows with only 0s are at the bottom, i.e. in the last row(s) 2) The first non-zero each row is a 1 --> this is called the "leading 1" 3) Leading 1s occur one place to the right of the leading 1 in the row above 4) Each column with a leading 1 has zeros elsewhere 30:22 Example 32:42 R2 --> 5R1+R2 & then R3 --> 2R1 + R3 34:50 Question: Why start with eliminating the number in the top right rather than the pivot (top left)? Answer: To avoid making fractions within the matrix. It's easier to work with whole numbers. 35:50 The order of the columns can technically be swapped. It would not change the system of equations. It would just been the order in which the variables appear in all of the equations would change. 36:20 No way to easily create a 1 without introducing fractions. We want to avoid altering R1 because otherwise it could reintroduce numbers where we worked to put 0s. R3 --> 2R2 + R3 to make bottom middle number a 0 37:50 R3 --> 1/37*R3 to get leading 1 in first column 38:24 R1 R3 rows can be switched to get 1 in the top right position of the matrix. This just changes the order or the equations (not the order of the variables) 39:24 R2 --> -17R1 + R2 40:24 R3 --> -3R1 + R3 41:34 R2 --> -1/2*R2 to get leading 1 in row 2 41:58 R3 --> R2+R3 42:26 Arrived at final solution 42:42 Understanding how time could be saved when solving using row echelon form. Once you find the solution for 1 variable, you can use subsitution to help find values for the other variables. Walkthrough of solving the system this way. 46:50 Question regarding what the numbers after vertical line mean in the matrix. Answer: it represents the constants which occurs after the equal sign in the equations. The vertical line in the matrix represents the equal sign. The matrix is known as an "Augmented coefficient matrix." 47:30 Generalisation i.e. general formula for system of linear equations, when you don't know how many equations & variables there are. Notation used is explained (i.e. subscripts are used to label the variables & coefficients) 51:20 (Augmented) coefficient matrix, using general formula 52: 24 Row operations i.e. the alternations that can be made to the matrix when reducing it (partially or fully) into row echelon form 1) Adding (or subtracting) a row (or a multiple of the row) to another row a.k.a. replace a row with a linear combination of rows. (Linear combination will be discussed later on) 2) Multiplying a row by a constant 3) Swapping rows (i.e. changing the order of the rows) 54:54 In practice, you generally only do the first type of row operation, and once you arrive at the solution for one variable, you can use back substitution to find the other variables 55:20 Example of solving an undermined system (where there are fewer equations than variables) by reducing it to row echelon form. In this example, there are 2 equations and 3 variables. Row echelon form appears a little differently in this case because there are more variables than equations. The final answer is written in terms of a parameter, t. 1:01:26 Homogenous System of Equations Homeogenous means that the equations come from the same family. It means that all constants (i.e. that number without a variable, which occurs after the equal sign) is 0. Therefore, all variables equal to 0 is a solution (e.g. x=0, y=0, z=0). However, there might be other solutions - the matrix could reduce down to the underdetermined system, which means there would be infinitely many solutions. So if there is a solution other than 0, then there are infinitely many solutions. 1:05:18 Example of solving a homogeneous system. When reduced to row echelon form, end up with the bottom row that has just 0s. The final answer is written in terms of the parameter, t. 1:11:26 Introduction to Matrices - general notation A matrix is a rectangular array of numbers. "Matrix" is singfular & "matrices" is plural. Matrices are typically denoted by capital letters e.g. A, B, X, Y 1:13:54 The general matrix (formula) 1:16:00 The size of a matrix is described as having "dimensions/order" of "m x n", where 'm' is the number of rows and 'n' is the number of columns 1:16:49 A 1 x n matrix is known as a "row vector" 1:17:50 A m x 1 matrix is known as a "column vector"

Here’s an outline of this lecture 1. Solving Linear System Using Matrices 1.1 Notation 1.2 Organise the Coefficients in a Matrix 2. Echelon form of a Matrix 2.1 Key features 2.2 Echelon Notation 2.3 Substitution Method 3. Generalisation 3.1 m Equation, n Variables 3.2 Augmented Coefficient Matrix 3.2.1 Row Operations 4. Underdetermined System 5.Homogenous System of Equations 6. Matrices 6.1 General Notation 6.2 Row & Column Vectors

Love the professor! This is how some one who loves his job looks...

Teacher:Is there any quistions? Student:🤐 Is it like this everywhere!!

The cameraman really does not want me reading that equation.

Like the professor a lot. Thanks for the free course

Where did they get this cameraman from? Was it a student receiving extra credit for filming AND taking notes?

big love from Saudi, beautiful lectures

It's many details but clear examples 😊🎉

At 45:00, he said "row 3" instead of "row 1". Row 1: z = -1.

the cameraman is drunk

learnt so much Lecturer Jason I'm so grateful! thanks for sharing !! :)

What a beautiful lecture!

wish the volume of all of these was higher. I have my youtube player and pc turned all the way up.

Very good lecture! Thank you OCW Irvine :)

Bros late to class at 28:08 lol. I feel sorry for them cause this shit is hard to grasp even from the start.

Poor camera focussing. Unable to follow the equation on the board. Cameraman running around the prof.

What is the use of so many key features of row echelon format?

Shouldn't they also provide with the homework assignments ( provided to the students of the class ) on there OCW website?

The job I want to do doesn’t even require college so I don’t even know if I should go?

At 43:00 it should be z y X order...

what is wrong with cameraman ?!! just zoom out a little and stay still on the board ...he moves the camera with every move of teacher

At 58", why not multiply by positive 1/3?

Is this class for an undergraduate degree?

31:00

cameraman rlly wants us to not learn this by not showing us half what what he's talking about

next time, stop flying kite with camera

23:00

Matrices are so based ong

16:55 haha he didn't answer/understand the question. And in case that student, three years later, is still looking for an answer: Yes, R(2)-R(1) = -( R(1) - R(2) ), and 0 = -(0)

dude this cameraman....

Maths. Not economics. Econ; dont need so much maths. Read, Capital by Marx. This shit is not there.