Lecture 1: Introduction to Linear Algebra by Gilbert Strang

Course Overview

• MIT Course: 18.06 Linear Algebra
• Text: "Introduction to Linear Algebra"
• Course webpage: mit.edu/18.06
• Includes exercises, MATLAB codes, and syllabus

Fundamental Problem of Linear Algebra

• Solve a system of linear equations.
• Focus on cases with equal numbers of equations and unknowns (n equations and n unknowns).

Visualizing Linear Equations

Row Picture

• Visual representation of equations one at a time.
• Example: Two equations with two unknowns.
  • Example equations:
    1. 2x - y = 0
    2. -x + 2y = 3
  • Coefficient matrix (A):
    | 2 -1 |
    | -1 2 |
  • Unknown vector (x):
    | x |
    | y |
  • Right-hand side vector (b):
    | 0 |
    | 3 |

Solving the Example

1. First Equation:

   • Find solutions for 2x - y = 0
   • Plot points: (0,0), (1,2)
   • Resulting line is all solutions to the equation.

2. Second Equation:

   • Find solutions for -x + 2y = 3
   • Example points: (0,3) and (-1,1).
   • Resulting line is drawn.

3. Intersection Point:

   • Verify intersection point (1,2) solves both equations.
   • Graphic representation of the solution.

Column Picture

• Analyze columns of the matrix A.
• The equation can be expressed as a linear combination of columns:
  A * x = b
• Goal: Find coefficients that produce the right-hand side vector b.
• Example of visualizing column combinations to produce the desired vector.

Transition to 3 Equations in 3 Unknowns

• Move from 2D to 3D examples to extend concepts.
• Example equations:
  1. 2x - y = 0
  2. -x + 2y - z = -1
  3. -3y - 3z = 4
• Understand equations geometrically: each represents a plane in 3D.
• Intersection of planes leads to potential solutions.

Concept of Linear Combinations

• Importance of understanding linear combinations of columns.
• Check if combinations fill the multidimensional space (e.g., 3D space).
• Concept of singular vs non-singular matrices and their implications on solution existence.

Matrix Representation and Multiplication

• Matrix form: A * x = b.
• Matrix multiplication:
  • Can be viewed as a combination of columns or a dot product of rows.
• Example of multiplying a matrix by a vector to illustrate concepts.

Summary

• Review of the key ideas presented in this lecture:
  • Importance of visualizing equations through row/column pictures.
  • The role of linear combinations in defining solutions.
  • The necessity of understanding the conditions under which solutions exist.
• Next lecture: Focus on systematic elimination methods for solving linear equations.