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Applications of linear systems
- Intro Lesson: a21:55
- Intro Lesson: b14:08
- Intro Lesson: c15:04
- Lesson: 126:06
- Lesson: 223:07
- Lesson: 317:31
Applications of linear systems
Lessons
We will be looking at real-life problems involving linear algebra. The three types of real-life applications we will be looking at are:
1. Linear systems in Economics
2. Linear systems with Chemical Equations
3. Linear systems with Network Flow
In economics, we can use linear algebra to determine the equilibrium price of outputs for each sector. Note that in order to get the equilibrium price, we need to set
Income = expenses (expenditures)
In Chemistry, we can use linear algebra to balance chemical equations like:N2+H2→NH3
We do so by counting the number of elements in a compound, and turning each coefficient as a variable to solve.
We can also use linear algebra to study the flow of some quantity through a network. Make sure that for each node:Flow in = Flow out
Our goal is to make all of these questions into matrix, and then solve.
1. Linear systems in Economics
2. Linear systems with Chemical Equations
3. Linear systems with Network Flow
In economics, we can use linear algebra to determine the equilibrium price of outputs for each sector. Note that in order to get the equilibrium price, we need to set
In Chemistry, we can use linear algebra to balance chemical equations like:
We do so by counting the number of elements in a compound, and turning each coefficient as a variable to solve.
We can also use linear algebra to study the flow of some quantity through a network. Make sure that for each node:
Our goal is to make all of these questions into matrix, and then solve.
- IntroductionApplications of Linear Systems Overview:a)Linear Systems in Economics
• A simple economy with sectors.
• Solving the equilibrium pricesb)Linear Systems with Chemical Equations
• Balancing chemical equations
• Solving for coefficients to balancec)Linear Systems with Network Flow
• Nodes
• Finding the general solution of the network flow - 1.Economics with Resources
Assume that the economy has 3 sectors: Manufacturing, Services, and Extraction. Manufacturing sells 50% of its output to Services, and 50% to Extraction. Services sells 30% of its output to Manufacturing, 60% to extraction, and keeps the rest. Extraction sells 80% of its output to manufacturing, 10% to services, and keeps the rest. Find a set of equilibrium prices for when the extraction output is 80 units. - 2.Balancing a Chemical Equation
Balance the following chemical equation using the vector equationsNa3PO4+MgCl2→NaCl+Mg3(PO4)2 - 3.Finding the General Flow Pattern
Find the general solution of the network flow. Assuming that all flows are non-negative, what is the minimum value of x1 and x2?
Do better in math today
1.
Linear Equations with Matrices
1.1
Notation of matrices
1.2
Solving systems of linear equations by graphing
1.3
Representing linear system as a matrix
1.4
The three types of matrix row operations
1.5
Solving a linear system with matrices using Gaussian elimination
1.6
Row reduction and echelon forms
1.7
Linear combination and vector equations in Rn
1.8
Matrix equation Ax=b
1.9
Solution sets of linear systems
1.10
Application of linear systems