Lecture 1: Introduction to Linear Algebra (MIT 18.06) - Gilbert Strang

Course Information

• Textbook: Introduction to Linear Algebra
• Course Webpage: web.mit.edu/18.06
• Focus: Solving systems of linear equations

Lecture Plan

• Fundamental problem: Solving systems of linear equations
• Types of representations:
  • Row picture
  • Column picture
  • Matrix form

Example Problem: 2x - y = 0 and -x + 2y = 3

• Matrix Form:

  • Coefficient matrix A:

    $$\begin{bmatrix} 2 & -1 \ -1 & 2 \end{bmatrix}$$

  • Unknowns vector x:

    $$\begin{bmatrix} x \ y \end{bmatrix}$$

  • Right-hand side vector b:

    $$\begin{bmatrix} 0 \ 3 \end{bmatrix}$$

• Equation: $A x = b$

Row Picture

• First Equation: 2x - y = 0
  • Points on the line: (0,0), (1,2), etc.
• Second Equation: -x + 2y = 3
  • Points on the line: (-3,0), (-1,1), etc.
• Solution: Intersection of the lines (1,2)

Column Picture

• Columns:

  • First column:

    $$\begin{bmatrix} 2 \ -1 \end{bmatrix}$$

  • Second column:

    $$\begin{bmatrix} -1 \ 2 \end{bmatrix}$$

• Objective: Find linear combination of columns to produce vector b:

  $$1 \cdot \begin{bmatrix} 2 \ -1 \end{bmatrix} + 2 \cdot \begin{bmatrix} -1 \ 2 \end{bmatrix} = \begin{bmatrix} 0 \ 3 \end{bmatrix}$$

• Key Idea: Linear combinations of columns

Bigger Picture: Any Number of Equations and Unknowns

• For larger systems, similar principles apply
• Example: 3x3 system of equations
• Solving involves finding where three planes intersect
• Column picture: Linear combinations of three vectors to produce a target vector

Important Concepts

• Linear Combinations: Combining vectors using scalar multiplication and addition
• Matrix Multiplication:
  • Column interpretation: Linear combinations of columns
  • Row interpretation: Dot products of rows with vector x

Questions to Consider

• Can every equation system be solved? Not always.
• Example of Non-Solution:
  • If three columns lie in the same plane, linear combinations can't cover all possible right-hand sides b
  • Matrix would be singular or not invertible

Higher Dimensions

• Conceptually, principles apply to higher dimensions (e.g., 9x9 systems)
• Linear combinations fill the n-dimensional space if vectors are independent

Conclusion

• Next lecture: Systematic method for solving equations (Elimination)

Key Takeaways

• Understanding the different pictures (row, column, matrix) is crucial in linear algebra
• Goal: Find if the system has a solution and if so, how to find it using various methods