MIT Course 18.06: Linear Algebra - Lecture 1

Instructor Introduction

• Gilbert Strang
• Textbook: Introduction to Linear Algebra
• Course website: MIT 18.06
• Lecture includes examples and practical applications.

Key Topics of Lecture 1

• Fundamental Problem of Linear Algebra: Solve a system of linear equations.
• Case Study: n equations and n unknowns (ideal scenario).

Visual Representations

1. Row Picture: Visualizing one equation at a time.

   • Example: 2x - y = 0 and -x + 2y = 3.
   • Graphs of equations represented as lines in the XY plane.
   • Important to identify intersection points as solutions.

2. Column Picture: Focus on matrix columns and their combinations.

   • The equations can be represented in matrix form: Ax = B.
   • Coefficients form the matrix A, unknowns form vector x, and right-hand side forms vector B.
   • Understanding linear combinations of matrix columns is essential.

Example of 2 Equations and 2 Unknowns

• Equations:
  1. 2x - y = 0
  2. -x + 2y = 3
• Coefficient Matrix (A):
  [ A = \begin{bmatrix} 2 & -1 \ -1 & 2 \end{bmatrix} ]
• Unknown Vector (x): [ x = \begin{bmatrix} x \ y \end{bmatrix} ]
• Right-hand Side Vector (B): [ B = \begin{bmatrix} 0 \ 3 \end{bmatrix} ]

Row Picture Example

• First Equation (2x - y = 0):

  • Points: (0, 0) and (1, 2) lie on the line.
  • Linear relationship indicates a straight line in the graph.

• Second Equation (-x + 2y = 3):

  • Identify points such as (0, 1.5) and (-3, 0).

• Intersection Point: (1, 2) is the solution for both equations.

Column Picture Example

• Linear Combination: Representing the equations using their columns to find B.
• Graphically combining columns effectively demonstrates how to achieve the right-hand side vector.

Generalization to 3 Equations and 3 Unknowns

• Transition to three dimensions (X, Y, Z).
• Each equation represents a plane in three-dimensional space.
• Understanding how these planes intersect is key to finding solutions.

Matrix Multiplication Understanding

• Matrix multiplication can be visualized as:
  • Column-wise: Combining columns of A scaled by components of vector x.
  • Row-wise: Using dot products of rows of A with vector x.
• Matrix Form: [ Ax = B ] where B is the desired output.

Conclusion and Next Steps

• Upcoming lecture focused on the method of elimination for solving systems of equations.
• Importance of identifying cases where solutions may or may not exist based on the properties of the matrix.