MIT Course 18.06 Linear Algebra Lecture 1 Notes

Introduction

• Instructor: Gilbert Strang
• Textbook: "Introduction to Linear Algebra"
• Course webpage: MIT 18.06
• Focus of the course: To solve systems of linear equations.

Fundamental Problem of Linear Algebra

• Solving systems of linear equations with n equations and n unknowns.

Pictures of Linear Equations

1. Row Picture
   • Visualizes one equation at a time (e.g., lines meeting in 2D for 2 equations).
   • Important: Check if they go through the origin.
2. Column Picture
   • Looks at the columns of the coefficient matrix.
   • Each equation represents a plane in higher dimensions.

Example: Two Equations, Two Unknowns

• Equations:
  • 1. 2x - y = 0
  • 2. -x + 2y = 3

• Coefficient Matrix (A):

  | 2 -1 | | -1 2 |
  • Dimensions: 2x2
• Unknowns:
  • Vector X = [x, y]
  • Right-hand side B = [0, 3]

Row Picture

• First Equation:
  • Points satisfying 2x - y = 0:
    • Origin (0,0)
    • Point (1, 2)
• Second Equation:
  • Points satisfying -x + 2y = 3:
    • Point (-3, 0)
    • Point (-1, 1)
• Intersection Point:
  • Solution (1, 2) lies on both lines.

Column Picture

• Represents the linear combination of column vectors to produce the right-hand side B.
• Key Concept: Linear Combination of columns.
  • E.g., Find coefficients to produce (0, 3):
    • 1 * Column 1 + 2 * Column 2 = (0, 3)

Three Equations, Three Unknowns

• Consider the equations in 3D space (x, y, z).
• Example Equations:
  • 1. 2x - y = 0
  • 2. -x + 2y - z = -1
  • 3. -3z + 2y = 4
• Each equation represents a plane in three-dimensional space.

Visualizing the Planes

• Planes intersect at lines or points; finding these intersections is crucial for understanding solutions.
• Column Picture:
  • Combing columns to produce the right-hand side.
  • Important to determine if three or more vectors (columns) fill the space.

Matrix Multiplication

• Matrix A times vector X results in linear combinations of the columns of A.
• Two methods to compute:
  • By columns: Combine the columns using the values in X.
  • By rows: Calculate using the dot product.

Conclusion

• The next lecture will cover systematic methods (elimination) to find solutions to systems of equations.
• Important concepts:
  • Linear combinations, geometry of equations, intersection of planes.