MIT Course 18.06 Linear Algebra - Lecture 1

Overview

• Instructor: Gilbert Strang
• Course Text: "Introduction to Linear Algebra"
• Course Webpage: MIT 18.06
• Lecture Focus: Fundamental problem of linear algebra - solving systems of linear equations.

Key Concepts

Systems of Linear Equations

• Focus on solving n equations with n unknowns.
• Different perspectives:
  • Row Picture: Visualizing one equation at a time.
  • Column Picture: Analyzing the columns of a matrix.
  • Matrix Form: Using matrices to represent equations (e.g., Ax = B).

Example: Two Equations, Two Unknowns

1. Equations:

   • 2x - y = 0
   • -x + 2y = 3

2. Coefficient Matrix (A):

   | 2 | -1 | | -1 | 2 |

3. Unknown Vector (X):

   | x | | y |

4. Right-Hand Side Vector (B):

   | 0 | | 3 |

Row Picture

• Plotting the equations in the XY plane.
• First equation (2x - y = 0):
  • Passes through the origin (0,0).
  • Example points: (0,0), (1,2).
• Second equation (-x + 2y = 3):
  • Does not pass through the origin.
  • Example points: (-3,0), (-1,1).
• Intersection (solution): Point (1,2).

Column Picture

• Analyzing the columns of the matrix:
  • Column 1: Vector (2, -1)
  • Column 2: Vector (-1, 2)
• Problem: Find coefficients for linear combination to achieve B.
• Example: 1
  • Combine column vectors to yield (0, 3).

Example: Three Equations, Three Unknowns

1. Equations:

   • 2x - y + 0z = 0
   • -x + 2y - z = -1
   • -3y - 3z = 4

2. Coefficient Matrix (A):

   | 2 | -1 | 0 | | -1 | 2 | -1 | | 0 | -3 | -3 |

3. Right-Hand Side Vector (B):

   | 0 | | -1 | | 4 |

Row Picture

• Three-dimensional space (X, Y, Z):
  • Each equation represents a plane.
  • Intersection of three planes gives point (solution).

Column Picture

• Combining columns to solve for B in three dimensions.
• Example illustration with various combinations of column vectors.

General Observations

• Linear Combinations: Fundamental operation in linear algebra.
• Can question whether every right-hand side B can be achieved:
  • Yes, if the columns span the space.
  • No, if columns are linearly dependent (e.g., all lie in the same plane).

Matrix-Vector Multiplication

• Two methods to multiply a matrix by a vector:

  • Column Method: Treat as linear combination of columns.
  • Row Method: Dot product of rows with the vector.

• Example given with matrix:

  | 2 | 5 | | 1 | 3 |

• Example calculation: A * X = B yields specific values.*

Next Steps

• Upcoming lecture will cover systematic elimination methods to find solutions for any system of equations and determine conditions for the existence of solutions.