Lecture 1: Introduction to Linear Algebra

Lecturer: Gilbert Strang

Course Information

• Course: MIT 18.06 Linear Algebra
• Textbook: Introduction to Linear Algebra
• Course Web Page: MIT 18.06
• Resources: Exercises, MATLAB codes, syllabus
• Video Lectures: TBA

Key Concepts of Lecture 1

Fundamental Problem of Linear Algebra

• Goal: Solve a system of linear equations
• Typical scenario: Equal number of equations and unknowns (e.g., $n$ equations and $n$ unknowns)

Approaches to Understanding Linear Equations

1. **Row Picture

   • Each equation represents a geometric line (2D case) or plane (3D case)
   • Example: $2x - y = 0$ and $-x + 2y = 3$
   • Matrix Form: Collect coefficients into a matrix $A$
   • System of equations in matrix form: $AX = B$

2. Column Picture

   • Focus on linear combinations of columns to form the right-hand side vector $B$
   • Example with $A$ matrix: $[[2, -1], [-1, 2]]$, unknowns $[x, y]^T$, and $B = [0, 3]^T$

3. Matrix Form

   • Understanding system $AX = B$
   • Matrix $A$: Coefficient matrix
   • Vector $X$: Vector of unknowns $(x, y)$
   • Vector $B$: Right-hand side vector $(0, 3)$

Solving the Example: 2 Equations, 2 Unknowns

• Matrix: $A = [[2, -1], [-1, 2]]$
• Solution: Plot lines representing each equation, find intersection point
• Row Picture: Intersection of lines $2x - y = 0$ and $-x + 2y = 3$ shows solution $(1, 2)$
• Column Picture: Combination: $[2, -1]^T + 2[-1, 2]^T = [0, 3]^T$

Moving to 3 Equations, 3 Unknowns

• Example: $2x - y = 0$, $-x + 2y - z = -1$, $-3y + 4z = 4$
• Matrix Representation: Expand approach to planes in 3D
• Example Matrix $A$ and $B$:

  A = [
  [2, -1, 0],
  [-1, 2, -1],
  [0, -3, 4]
  ]
  B = [0, -1, 4]
  

• Row Picture: Intersection of three planes
• Column Picture: Combination of vectors $[2, -1, 0]^T$, $[-1, 2, -1]^T$, $[0, -3, 4]^T$ to achieve $B$

General Observations

• Linear Combination: Fundamental operation (combination of columns to form vector $B$)
• Solving Ax = B: Dependent on the matrix $A$ and vector $B$
• Invertibility: Matrix must be non-singular (invertible) for every $B$ to have a unique solution
• Dependence and Independence: If columns of $A$ are linearly independent, every $B$ can be formed

Beyond 3 Equations: Higher Dimensions

• Concept: Extend intuition to 9 equations in 9 unknowns (9D space)
• Random Matrices: Generally non-singular, easy with software like MATLAB
• Dependent Columns: Limitation where columns lie within the same plane

Example of Matrix Multiplication

1. Matrix-Vector Multiplication:

• Matrix $A$: $[[2, 5], [1, 3]]$
• Vector $X$: $[1, 2]^T$
• Result: $AX = [12, 7]^T$

2. Dot Product: Alternative method

Conclusion

• Next Steps: Systematic way of solving equations through elimination in the next lecture

Important Terms

• Row Picture
• Column Picture
• Matrix Form
• Linear Combination
• Invertible (Non-Singular) Matrix