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:
- 2x - y = 0
- -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:
- 2x - y = 0
- -x + 2y - z = -1
- -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!