Introduction to Linear Algebra Concepts

Aug 4, 2024

MIT Course 18.06 Linear Algebra - Lecture 1 Notes

Instructor: Gilbert Strang
Course Text: Introduction to Linear Algebra
Course Web Page: MIT 18.06


Overview

  • Fundamental Problem of Linear Algebra: Solve systems of linear equations.
  • Focus on systems with an equal number of equations and unknowns (n equations, n unknowns).
  • Introduce different perspectives:
    • Row Picture: Solving one equation at a time (graphical representation).
    • Column Picture: Viewing the problem through the columns of a matrix.
    • Matrix Form: Representing equations in matrix notation.

Example 1: Two Equations, Two Unknowns

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

Coefficient Matrix

[ A = \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix} ]

  • Vector of Unknowns: ( X = \begin{pmatrix} x \\ y \end{pmatrix} )
  • Right-Hand Side Vector: ( B = \begin{pmatrix} 0 \\ 3 \end{pmatrix} )

Row Picture

  • First Equation:
    • Points satisfying 2x - y = 0:
      • Origin (0,0) is a solution.
      • Example point: (1, 2).
  • Second Equation:
    • Points satisfying -x + 2y = 3:
      • If y = 0, then x = -3.
      • If x = -1, then y = 1.
  • Solution Point: (1, 2) lies on both lines.

Column Picture

  • Linear Combination: Representing the equation as a combination of column vectors.
  • Goal: Find coefficients x and y such that ( ax + by = B ).
  • Visualization of Columns:
    • Column 1: (2, -1)
    • Column 2: (-1, 2)
  • Combination resulting in B (0, 3):
    • ( x = 1, y = 2 ) gives the correct output.

Example 2: Three Equations, Three Unknowns

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

Row Picture

  • Understanding Planes:
    • Each equation represents a plane in 3D space.
    • The intersection of planes gives the solution.
  • Visualizing Three Planes Meeting:
    • Goal is to find the point of intersection of these three planes, which represents the solution.

Column Picture

  • Combining Columns:
  • Similar to the previous example, identify coefficients for each column vector to match the right-hand side vector.

General Questions

  • Can we solve ( Ax = B ) for every right-hand side B?
    • Depends on the independence of the columns of A.
    • If columns are not independent, certain B's may not be reachable.

Matrix Multiplication

  • Matrix-Vector Product:
  • Can be approached through:
    • Column View: Linear combination of columns.
    • Row View: Dot product of rows with the vector.

Future Topics

  • The next lecture will cover elimination methods to systematically find solutions to the systems of equations.

Conclusion:

  • Focus on understanding the geometry behind the equations and how linear combinations work to produce solutions.
  • Important concepts to revisit include row and column pictures, the significance of linear combinations, and matrix multiplication.

Thank you for attending!