Systems of Equations Quiz

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Linear Algebra - Week 1

• Course topic: Math for Machine Learning
• Subtopic: Linear algebra

System of Linear Equations

• Subtopic: Linear Algebra Applied I

Machine Learning

• Depicts graphical representations of machine learning concept
• Includes diagrams of neural networks, data points on a graph, and document icons.

Machine Learning Specialization

• Advertisement for DeepLearning.AI's Machine Learning Specialization course
• Includes an interactive robot character and human instructors, along with call to enroll.

Linear Algebra and Machine Learning

• Topic: Linear Regression
• Subtopic: Supervised Machine Learning
• Input data flows into the process
• Output is generated from the input
• Wind speed is cited as an input example
• Electrical power output is an example of an output

Linear Algebra and Machine Learning

• Input: Wind speed
• Output: Power output

Linear Algebra and Machine Learning

• Input-output relationship illustrated on a graph displaying a plot of wind speed against power.
• The plot is a line representing a linear regression model.

Linear Algebra and Machine Learning

• Input/output example illustrated with wind speed and power output.

Linear Algebra and Machine Learning

• Input features mentioned: wind speed, temperature
• Output example: power output

Linear Algebra and Machine Learning

• Input features: wind speed, temperature, and pressure, humidity
• Output feature: power output

Linear Algebra and Machine Learning

• Features are introduced: wind speed, temperature, pressure, humidity, and other features
• Output/target output
• Formula introduced: y=Wx +b

Linear Algebra and Machine Learning

• Input features (variables): wind speed, temperature, pressure, humidity, others
• Output/target: power output

Linear Algebra and Machine Learning

• A system of linear equations is presented in a matrix format.
• This shows the relationship between multiple inputs and a single output or target.

Linear Algebra Applied II - System of Linear equations

Linear Algebra and Machine Learning

• Illustrates graphical relationship
• Input/output relationship using a graph

Linear Algebra and Machine Learning

• The concepts of input and output is further developed.
• Features such as wind speed and temperature are described alongside output quantity, power output.

Linear Algebra and Machine Learning

• Input features and their types are explicitly shown
• Features: wind speed, temperature, pressure, humidity, other features
• Target/output: Power ouput

Linear Algebra and Machine Learning

• A matrix is presented showing the mathematical relationship between variables and their output/target

Linear Algebra and Machine Learning

• System of linear equations formalized as an algebraic series of equations

System of Linear Equations

• Plan for the week
• Includes subtopics: common vector and matrix operations; Questions (to-do list), end of week

Plan for the Week

• Subtopics: systems of linear equations;
• representing systems as vectors and matrices, computing determinants

Check your Knowledge

• The problem involves three statements as equations.
• Statements relating linear algebra, calculus, and probability scores
• Mathematical representations of real-world information are presented

Check your Knowledge

• Questions regarding weights, features, and targets in a linear equation system
• Values for the targets are provided.

Check your Knowledge

• Is the system singular or non-singular?
• Questions: solving the system of equations, matrix representation, calculation of determinant

What to Expect

• Indicates the steps and learning stages to complete.
• Lists steps in the learning process

Systems of Sentences

• System 1: complete and non-singular
• System 2: redundant and singular
• System 3: contradictory and singular

Systems of Sentences

• Examples of different systems of sentences
• Includes complete, redundant, and contradictory sentence systems

Quiz: Systems of Sentences

• The quiz question has statements describing relationships between the dog, cat and a bird(or other objects).
• Questions regarding the color of the bird and singularity of the system.

Solution: Systems of Information

• Finding the color of the bird
• The system is determined as being non-singular

System of Equations

• Converting sentences to equations through numerical representation.
• Example: the price of an apple and a banana cost $10

Quiz: Systems of Equations 1

• Statements regarding buying fruits
• The quiz requires calculating the price of each for the fruits purchased

Solution: Systems of Equations 1

• The solution represents the calculation for the fruit prices based on the provided information

Quiz: Systems of Equations 2

• Real-world problem
• The quiz statement describes purchasing different combinations of fruits (apples, bananas, and cherries) at different costs

Solution: Systems of Equations 2

• Showing the systems of linear equations related to the problem

Quiz: Systems of Equations 3

• Statements regarding apples and bananas prices based on purchasing scenarios
• The price of each fruit(apple, banana) needs to be calculated

Solution: Systems of Equations 3

• Solution based on the provided data for prices of apples and bananas.

Quiz: Systems of Equations 4

• Statements show buying scenarios with fruits (apple, banana cost)
• The quiz task for calculating the fruit cost.

Solution: Systems of Equations 4

• Statements about the buy scenarios for fruits (apple, banana)
• The solution is a contradiction and no solution is possible

Systems of Equations

• A set of equations (variables: a, b) are presented.
• Each system is analyzed and a proper classification is made based on multiple properties.
• The types presented include unique solution, infinite solutions, and no solution

Quiz: More Systems of Equations

• Three systems of equations are described, each containing three variables (a, b, and c).
• Each system of linear equations is to be properly solved and classified

Solutions: More Systems of Equations

• Solutions to the three systems of equations are shown and classified
• Examples include infinitely many solutions, no solutions, and infinite solutions

What is a Linear Equation?

• Distinguishes linear and non-linear equations
• Example equations from both categories illustrated.

System of Linear Equations

• Describes systems of equations as lines and planes

Linear equation → line

• Represents linear equations graphically as lines on a coordinate plane
• Illustrates unique solutions from the intersection of lines

Linear equation → line

• Representation of linear equations as a set of lines or a plane within a 3d coordinate

Linear equation → line

• Unique graphical representation of linear equations as lines

Linear equation → line

• Graphical representations of linear equations as lines, including possible outcomes (unique solution)

Systems of equations as lines

• Linear equations as lines on a coordinate plane
• Different possible outcomes presented (unique solution, infinite solutions, no solutions)

Quiz

• The problem involves plotting different systems of equations

Solution

• The problem asks which plot is the best fit for two linear equations

Linear equation in 3 variables as a plane

• The geometrical interpretation of a linear equation in three variables

Linear equation in 3 variables as a plane

• Linear equation in three variables illustrated

System 1

• Set of equations are presented
• Further geometrical description using diagram

System 2

• Equations shown
• Illustration shown

System 3

• Equations shown
• Illustrations shown

System of Linear Equations

• Discusses the concept of singularity.

Systems of equations as lines

• Explains systems of linear equations as lines or planes
• Shows different scenarios (unique solution, infinite solutions, no solutions)

Systems of equations as lines

• Explains how systems of linear equations can represent lines

Systems of equations as matrices

• Explains systems of equations as matrices with singular and non-singular matrices given.

Constants don't matter for singularity

• Explains that the constant values do no affect the singular nature of a system

Constants don't matter for singularity

• Describes how constant values in a series of equations do not affect the singular or non-singular nature of a matrix

Constants don't matter for singularity

• Explains the properties of linear equations, including singular and non-singular properties

Quiz: Determinants

• Determines determinants for different matrices

Solution: Determinants

• Solutions for the previously mentioned quiz on calculating determinants for different matrices.

The determinant

• Illustration of the determinant in a 3x3 matrix.
• Illustrates calculations for different diagonals of the matrix.

Conclusion

• Summary of the main concepts covered in the study material.