# M4ML - Linear Algebra - 1.2 Motivations for linear algebra

TLDRThis video introduces two key problems in Linear Algebra: price discovery through solving simultaneous equations, using the example of purchasing apples and bananas, and the optimization challenge of fitting data with an equation. It explains how these problems can be approached using computer algorithms and matrices, setting the stage for a deeper exploration of vectors, matrices, and their applications in future modules, including their relevance to neural networks and machine learning.

### Takeaways

- 🔢 Linear Algebra is a tool that helps solve problems involving simultaneous equations and data fitting.
- 🍎 The 'apples and bananas' problem illustrates the concept of price discovery through solving simultaneous equations.
- 🛒 Two shopping scenarios with different quantities of apples and bananas are used to establish a system of linear equations.
- 💶 The cost of items in the examples is represented by the variables A for apples and B for bananas.
- 🔍 To find the individual prices (A and B), one must solve the system of linear equations, which can be challenging for numerous items and shopping trips.
- 💻 A computer algorithm can efficiently solve such systems of equations, which is useful in the general case.
- 📊 The script also introduces the problem of fitting an equation to a set of data, such as a histogram representing a population.
- 🎯 The goal in data fitting is to find the optimal parameters that best describe the data, like finding the line that best fits a histogram.
- 📈 Linear Algebra involves working with vectors and matrices, which are central to solving the types of problems discussed.
- 📚 The video outlines a course structure that will cover understanding vectors and matrices, and then solving the presented problems.
- 🔑 The key takeaway is that Linear Algebra is essential for problems in optimization, price discovery, and data analysis.

### Q & A

### What is the main topic of the video?

-The main topic of the video is an introduction to Linear Algebra and the types of problems it can help solve.

### What is the first problem discussed in the video?

-The first problem discussed is price discovery, specifically determining the individual prices of apples and bananas based on two shopping trips.

### How many apples and bananas were bought in the first shopping trip?

-In the first shopping trip, two apples and three bananas were bought.

### What was the total cost of the items bought in the first shopping trip?

-The total cost of the items in the first shopping trip was eight Euros.

### What does the term 'simultaneous equations' refer to in the context of the video?

-In the context of the video, 'simultaneous equations' refers to a set of linear equations that are solved together to find the unknown values, in this case, the prices of apples and bananas.

### Why might one need a computer algorithm to solve simultaneous equations in a general case?

-A computer algorithm might be needed to solve simultaneous equations in a general case because it could involve a large number of items and shopping trips, making it difficult to solve by hand.

### What is the second problem discussed in the video related to Linear Algebra?

-The second problem discussed is the optimization problem of fitting an equation to data, which is relevant in fields like neural networks and machine learning.

### What is the purpose of fitting an equation to data?

-Fitting an equation to data allows for a compact representation of the data, such as describing a population's characteristics without needing all the original data, which can be useful for privacy reasons.

### What is a histogram in the context of the video?

-In the context of the video, a histogram represents a set of data visually, showing the distribution of a population with its average and variation.

### What is the goal when fitting an equation to a histogram?

-The goal is to find the optimal values of the parameters in the equation that best fit the data, providing the best description of the population distribution.

### What are the two main problems with Linear Algebra discussed in the video?

-The two main problems discussed are: 1) the problem of apples and bananas, which involves solving simultaneous equations, and 2) the optimization problem of fitting data with an equation that has fitting parameters.

### Outlines

### 🍎 Introduction to Linear Algebra and Problem Solving

This paragraph introduces the video's focus on exploring problems that can be addressed with Linear Algebra. It presents a real-world scenario of price discovery for apples and bananas, using simultaneous equations to determine the individual prices. The paragraph emphasizes the complexity of solving such equations by hand for numerous items and shopping trips, suggesting the utility of computer algorithms for these general cases. The concept of linear coefficients and how they relate to input variables (A and B) and output values (8 and 13 Euros) is explained. The paragraph sets the stage for the video series by outlining the intention to cover vectors, matrices, and methods to solve general linear algebra problems, including those relevant to machine learning and neural networks.

### Mindmap

### Keywords

### 💡Linear Algebra

### 💡Price Discovery

### 💡Simultaneous Equations

### 💡Vector

### 💡Matrix

### 💡Optimization

### 💡Histogram

### 💡Parameters

### 💡Machine Learning

### 💡Neural Networks

### 💡Multivariate Calculus

### Highlights

The video discusses the application of Linear Algebra in solving real-world problems.

The first problem introduced is price discovery through simultaneous equations.

An example is given where the cost of apples and bananas is determined using two shopping trips' data.

The prices of apples (A) and bananas (B) are unknowns in the linear equations.

The general case of multiple items and shopping trips makes manual solution difficult, suggesting the need for computer algorithms.

The video presents a Linear Algebra problem with constant linear coefficients relating input variables to output values.

The concept of vectors and matrices is introduced as part of the mathematical framework for solving these problems.

Modules one to three will focus on understanding vectors and matrices and how to work with them.

The second problem discussed is fitting an equation to data, relevant in fields like machine learning and neural networks.

A histogram representing a population with an average and variation is used as an example of data to be fitted.

The goal is to find the optimal parameters for the equation that best fits the data.

Using the fitted equation allows for an easy, portable description of the population without the need for original data.

The video mentions that the next module will look at how to measure the quality of the fit in terms of parameters.

Two main problems are set up in the first module: solving simultaneous equations and optimization for data fitting.

These problems will be revisited throughout the course on Linear Algebra and its application in multivariate calculus.

The video aims to expose what Linear Algebra is and how it can help solve various types of problems.

The practical applications of Linear Algebra include shopping price discovery and data fitting in machine learning.

The course will build up from basic concepts to more complex problem-solving techniques.