Week 2
Matrices for solving systems of equations

Business

The start of the week email. No need to reread it if you received it. It's just linked here in case anyone accidentally deleted it (it's happened before 🥺). Definitely let me know if you didn't receive it though so we can figure out why not.

[Optional] Just in time review

A couple of things that we'll use this week in class or on the assignment. Just in case you've forgotten.😉

⦿ Monday: Introducing matrices and matrix operations

Matrices: Terminology and examples Slide Show
If we're going to work with matrices, we need a way to talk about them, and that requires a bit of terminology. Terms are defined here that I will use going forward.
The slide show also gives several examples of applications of matrices to give you an idea of the wide variety of things they are used to model.
A brief history of matrices ↗︎
A nice brief illustrated history of matrices. Read a comic to learn math! 😎
Example: Solving a 2x2 system of equations using matrices Pencast 7:31|PDF
Side-by-side solution of the Pet Biscuits problem using matrix notation and algebraic notation.
Allowable matrix transformations Pencast 5:12|PDF
The transformations you can perform on a particular matrix depend on what the matrix represents. Since our matrices represent systems of equations we can do anything to the rows of our matrices that you could do to equations:
  1. Multiply all the entries in a row by a number
  2. Add (or subtract) one row from another
  3. Combining 1 and 2: Add (or subtract) any multiple of one row from another.
  4. Interchange rows.
  5. Interchange columns (but not the rightmost one).
Gaussian elimination (Pivoting) Pencast 10:52|PDF
It's time to explicitly identify the steps we go through to solve a system of equations using a matrix representation.
  1. Pivot around each element on the main diagonal of the matrix:
    1. Make the pivot element a 1,
      by dividing
      all the entries in the row by the pivot element.
    2. Make the other entries in the pivot column 0s,
      by adding/subtracting
      a suitable multiple of the pivot row from each other row.
  2. Read the solution from the final matrix.

Exercises

Time to practice:

  1. Writing the matrix corresponding to a system of equations.
  2. Writing the system of equations corresponding to a matrix.
  3. Dividing a matrix row by a number.
  4. Adding a multiple of one row to another.
  5. Pivoting a matrix about an entry.

Exercise set 2.1 Problems and Solutions.

⦿ Wednesday. All models are wrong

All models are wrong [Slide Show]
They are. Really. But that's not the whole story...

⦿ Wednesday: Solving systems of equations using matrices

Solving a 3x3 system using matrices: Example 1 Pencast 15:30|PDF
A step-by-step example of solving a 3x3 system of simultaneous equations using Gaussian elimination in a matrix.
Solving a 3x3 system using matrices: Example 2 Pencast 12:02|PDF
A second example of solving a 3x3 system of simultaneous equations using Gaussian elimination in a matrix.
Summary: Solving systems of linear equations using Gaussian elimination in matrices PDF
The title is quite the mouthful, but this is my capsule summary of the steps.

Exercises

Now we should be ready to put all the steps together to solve systems of equations using matrices.

Exercise set 2.2 Problems and solutions.

⦿ Friday: The computer helps out

Teaching Excel how to solve 3x3 systems of equations for us Screencast 17:34
Pivoting is fun and all, but it is a mechanical process, which means it could be automated, and done by a computer. Let's do that.

Exercises

Exercise set 2.3 Create your own Excel sheet to solve systems of equations. (It could be helpful in completing Assignment 2.)

All the exercises in one place

Exercise Set 2.1: Problems and Solutions. Due by 10:30 am Monday January 16.
Exercise Set 2.2: Problems and Solutions. Due by 10:30 am Monday January 16.
Exercise Set 2.3: Problems. Does not need to be submitted.

Assignment

Assignment 2. Due by 1 pm Friday January 19.

Remember if you get stuck don't hesitate to email me or see me after class.