This document provides an overview of lesson 12 on rank and solutions to systems of linear equations. It defines rank as the maximum number of linearly independent column vectors in a matrix and discusses how to compute rank using Gaussian elimination and minors. It also relates rank to the consistency and redundancy of systems of linear equations, noting that a system is consistent if the rank of the coefficient matrix equals the rank of the augmented matrix, and redundant or free variables exist if the ranks are less than the number of equations or variables respectively.

1. Lesson 12 (Sections 14.2–3)
Rank and Solutions to Systems
Math 20
October 19, 2007
Announcements
Midterm not graded yet.
Problem Set 5 is on the WS. Due October 24
OH: Mondays 1–2, Tuesdays 3–4, Wednesdays 1–3 (SC 323)
Prob. Sess.: Sundays 6–7 (SC B-10), Tuesdays 1–2 (SC 116)

2. Summary of Last time
The linear independence of a set measures its redundancy.

3. Deciding linear dependence
We showed
a1 , . . . , an LD ⇐⇒ c1 a1 + · · · + cn an = 0 has a nonzero sol’n

c1
.
⇐⇒ a1 . . . an  .  = 0 has a nonzero sol’n
.
cn
A
c
⇐⇒ system has some free variables
⇐⇒ rref(A) has a column with no leading entry to it

4. Deciding linear independence
So
a1 , . . . , an LI ⇐⇒ every column of rref(A) has a leading entry to it
In
⇐⇒ A ∼
O

5. Relation to invertibility
Let A be an n × n matrix. If A has an inverse A−1 , the only
solution to Ac = 0 is the zero solution.

6. Relation to invertibility
Let A be an n × n matrix. If A has an inverse A−1 , the only
solution to Ac = 0 is the zero solution.
This means that there is no linear dependence relation among the
columns.

7. Relation to invertibility
Let A be an n × n matrix. If A has an inverse A−1 , the only
solution to Ac = 0 is the zero solution.
This means that there is no linear dependence relation among the
columns.
Fact
A is invertible if and only if the columns of A are linearly
independent,

8. Relation to invertibility
Let A be an n × n matrix. If A has an inverse A−1 , the only
solution to Ac = 0 is the zero solution.
This means that there is no linear dependence relation among the
columns.
Fact
A is invertible if and only if the columns of A are linearly
independent, if and only if rref(A) = I.

9. Worksheet

10. Example
Solve
x+2y +z =1
2x+2y =1
x+3y +z =1

11. Example
Solve
x+2y +z =1
2x+2y =1
x+3y +z =1
Solution
Since    
1211 100 1/2
1 1 0 0 0 1 0 0
1311 001 1/2
we have x = 1/2, y = 0, z = 1/2.

12. Example
Solve
x+2y − z =1
2x+2y =1
x+3y −2z =1

13. Example
Solve
x+2y − z =1
2x+2y =1
x+3y −2z =1
Solution
Since    
1 2 −1 1 10 10
 0 1 −1 0 
1 2 0 0
1 3 −2 1 00 01
we have no solution.

14. Example
Solve
x+2y − z =3
2x+2y =4
x+3y −2z =4

15. Example
Solve
x+2y − z =3
2x+2y =4
x+3y −2z =4
Solution
Since    
1 2 −1 3 10 11
 0 1 −1 1 
1 2 0 4
1 3 −2 4 00 00
The system is equivalent to x = 1 − z, y = 1 + z, where z is free.

16. Example
Solve
x+2y −3z =1
2x+4y −6z =1
3+6y −9z =1

17. Example
Solve
x+2y −3z =1
2x+4y −6z =1
3+6y −9z =1
Solution
Since    
1 2 −3 1 1 2 −3 0
 2 4 −6 1  0 0 0 1
3 6 −9 1 00 00
there is no solution.

18. The rank
Definition
The rank of a matrix A, written r (A) is the maximum number of
linearly independent column vectors in A.

19. The rank
Definition
The rank of a matrix A, written r (A) is the maximum number of
linearly independent column vectors in A. If A is a zero matrix, we
say r (A) = 0.

20. Computing the rank by Gaussian Elimination
Fact
If A and B are row equivalent (we can get from one to another by
row operations), then r (A) = r (B).

21. Computing the rank by Gaussian Elimination
Fact
If A and B are row equivalent (we can get from one to another by
row operations), then r (A) = r (B).
So the rank of a matrix is equal to the rank of its RREF, which is
easy to calculate.

22. Example
Compute the ranks of the matrices
     
1 2 −1 1 2 −3
1 21
2 4 −6
2 2 1 2 2 0 
1 3 −2 3 6 −9
1 31

23. Example
Compute the ranks of the matrices
     
1 2 −1 1 2 −3
1 21
2 4 −6
2 2 1 2 2 0 
1 3 −2 3 6 −9
1 31
Answer.
3, 2, and 1.

24. Computing the rank by minors
Fact
The rank r (A) of a matrix is equal to the order of the largest
minor of A which has nonzero determinant.

25. Computing the rank by minors
Fact
The rank r (A) of a matrix is equal to the order of the largest
minor of A which has nonzero determinant.
This is not an obvious fact, nor is it easy to prove.

26. Rank and consistency
Fact
Let A be an m × n matrix, b an n × 1 vector, and Ab the matrix A
augmented by b.

27. Rank and consistency
Fact
Let A be an m × n matrix, b an n × 1 vector, and Ab the matrix A
augmented by b.
Then the system of linear equations Ax = b has a solution (is
consistent) if and only if r (A) = r (Ab ).

28. Rank and redundancy
Fact
Let A be an m × n matrix, b an n × 1 vector, and Ab the matrix A
augmented by b. Suppose that r (A) = r (Ab ) = k < m (m is the
number of equations in the system Ax = b).

29. Rank and redundancy
Fact
Let A be an m × n matrix, b an n × 1 vector, and Ab the matrix A
augmented by b. Suppose that r (A) = r (Ab ) = k < m (m is the
number of equations in the system Ax = b).
Then m − k of the equations are redundant; they can be removed
and the system has the same solutions.

30. Rank and redundancy
Fact
Let A be an m × n matrix, b an n × 1 vector, and Ab the matrix A
augmented by b. Suppose that r (A) = r (Ab ) = k < n (n is the
number of variables in the system Ax = b).

31. Rank and redundancy
Fact
Let A be an m × n matrix, b an n × 1 vector, and Ab the matrix A
augmented by b. Suppose that r (A) = r (Ab ) = k < n (n is the
number of variables in the system Ax = b).
Then n − k of the variables are free; they can be chosen at will and
the rest of the variables depend on them, getting infinitely many
solutions.