Lesson Notes By Weeks and Term v4 - SHS 2

Lesson Video

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Performance objectives

• Transform linear equations into matrix form (and vice versa).
• Describe the components of a matrix.
• Solve real-world problems by setting up matrices.

Lesson summary

This lesson introduces a powerful method for organising and representing complex problems. We often encounter situations in life with multiple unknown quantities and multiple conditions, from calculating business profits to managing resources. A system of linear equations is one way to describe these situations. However, as the problems get bigger (more unknowns), this format can become messy. Today, we will learn how to translate these systems of equations into the neat, organised structure of matrices. This skill is the first step towards using powerful mathematical tools (and even computers) to solve real-world problems efficiently in fields like economics, engineering, and data science.

Lesson notes

What is a System of Linear Equations?

A system of linear equations is a set of two or more linear equations with the same variables. Our goal is usually to find the values of the variables that satisfy all equations simultaneously. Example (2 variables): `2x + 5y = 12` `x - 3y = -1` Example (3 variables): `x + y + z = 6` `2x - y + 3z = 9` `x - 2y - z = -2` The Matrix Form: AX = B

Any system of linear equations can be written in a compact matrix form: AX = B. Let's break down what each part represents: A - The Coefficient Matrix: This is a matrix containing all the numerical coefficients (the numbers in front of the variables) from the left-hand side of the equations. The coefficients must be arranged in the same order as they appear in the equations. If a variable is missing in an equation, its coefficient is 0. X - The Variable Matrix: This is a column matrix (a matrix with only one column) that lists all the variables in the system. B - The Constant Matrix: This is a column matrix that contains all the constant terms from the right-hand side of the equations.

The Logic Behind AX = B The form AX = B works because of the rules of matrix multiplication. When you multiply the coefficient matrix A by the variable matrix X, the result is a column matrix where each entry corresponds to the left-hand side of one of the original equations.

Evaluation guide

• Transform systems of linear equations into a Matrix form and state the matrix representing
• a linear transformation.
• Collaborative Learning, Experiential Learning, Problem -based Learning, Project -based
• Learning and Talk for Learning