Solving Ax Equals Zero Pivot Variables | Generated by AI
This video is Lecture 7 from MIT’s 18.06 Linear Algebra course (Spring 2005), taught by Professor Gilbert Strang. The topic is:
“Solving \( A\mathbf{x} = \mathbf{0} \): Pivot Variables and Special Solutions”
Key Concepts Covered in the Lecture:
- Homogeneous Systems (\( A\mathbf{x} = \mathbf{0} \))
- A system of linear equations where the right-hand side is the zero vector.
- Always has at least the trivial solution \( \mathbf{x} = \mathbf{0} \).
- If there are free variables, there are infinitely many solutions.
- Pivot Variables vs. Free Variables
- Pivot variables: Correspond to columns with pivots (nonzero leading entries) in the row reduced echelon form (RREF) of \( A \).
- Free variables: Correspond to columns without pivots (can take any value).
- The number of free variables = number of columns − rank of \( A \).
- Special Solutions (Basis for the Nullspace)
- For each free variable, set it to 1 and the others to 0, then solve for the pivot variables.
- These solutions form a basis for the nullspace of \( A \) (all solutions to \( A\mathbf{x} = \mathbf{0} \)).
- The nullspace is a subspace of \( \mathbb{R}^n \).
- Rank and the Nullspace
- If \( A \) is an \( m \times n \) matrix with rank \( r \):
- Number of pivot variables = \( r \).
- Number of free variables = \( n - r \).
- Dimension of the nullspace = \( n - r \).
- If \( A \) is an \( m \times n \) matrix with rank \( r \):
- Example Walkthrough
- Strang works through an example (likely a matrix with rank < number of columns) to illustrate:
- Finding the RREF.
- Identifying pivot and free variables.
- Constructing special solutions.
- Expressing the general solution as a linear combination of special solutions.
- Strang works through an example (likely a matrix with rank < number of columns) to illustrate:
- Geometric Interpretation
- The nullspace represents all vectors \( \mathbf{x} \) that are “collapsed” to zero when multiplied by \( A \).
- For a 3×3 matrix with rank 2, the nullspace is a line through the origin.
Why This Lecture Matters:
- Introduces the nullspace (a fundamental subspace in linear algebra).
- Connects row reduction (elimination) to the structure of solutions.
- Lays the groundwork for understanding linear independence, bases, and dimension in later lectures.
- Essential for solving systems, understanding transformations, and applications like differential equations or machine learning.
Where to Find More:
- Full course materials: MIT OCW 18.06
- Textbook: Strang’s “Introduction to Linear Algebra” (Chapter 2 covers this topic).
- Follow-up lectures: Next lectures discuss solving \( A\mathbf{x} = \mathbf{b} \) (non-homogeneous systems) and the column space.
Summary in One Sentence:
This lecture teaches how to find all solutions to \( A\mathbf{x} = \mathbf{0} \) by identifying pivot/free variables and constructing a basis for the nullspace using special solutions.