Cogito
Linear Algebra · Chapter 6 · Lesson 3
Rank and the Rank-Nullity Theorem
What survives plus what is crushed is always the whole input.
12 problems · about 24 minutes · N-VM.C.10, A-REI.C.8
What this lesson teaches
The student computes rank and nullity and applies the rank-nullity theorem.
- Rank is how many independent directions survive; nullity is how many are crushed.
- Rank plus nullity always equals the number of columns.
- A square matrix of full rank has nullity zero and is therefore invertible.
Warm Up
Straightforward practice. Get the method working first.
5 problems8 columns and rank 5. What is the nullity?
Answer 3
Why 3.
What does the rank-nullity theorem say?
Answer Rank plus nullity equals the number of columns.
Why They sum to the column count.
6 columns and rank 2. What is the nullity?
Answer 4
Why 6 − 2.
4 columns and nullity 1. What is the rank?
Answer 3
Why 4 − 1.
A 2 by 2 matrix with determinant 0 and non-zero entries. What is its rank?
Answer 1
Why One column is a multiple of the other.
Build It Up
The same ideas with more to keep track of.
3 problemsThe zero matrix with 3 columns. What is its rank?
Answer 0
Why Nothing survives.
The zero matrix with 3 columns. What is its nullity?
Answer 3
Why Everything is crushed.
A 3 by 3 identity matrix. What is its rank?
Answer 3
Why Full rank.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each quantity to what it measures.
Answer Rank → Independent directions that survive; Nullity → Independent directions crushed to zero; Rank plus nullity → The number of columns
Why The two parts must account for every column.
10 columns and rank 10. What is the nullity?
Answer 0
Why Full rank crushes nothing.
The Balance: A matrix with 7 columns and rank 4. What is its nullity?
Answer 3
Why 3.
The Full One: An invertible 5 by 5 matrix. What is its nullity?
Answer 0
Why 0.