Understanding Rank and Nullity of a Linear Transformation – Linear Algebra I, Section 5.2
Welcome, students! 😊 In this lesson, we will explore one of the most powerful ideas in linear algebra: rank and nullity of a linear transformation. Don’t worry if these words sound new — by the end of this guide, you’ll understand them deeply, with clear examples and practice. Let’s begin our journey together! 🙌
What Is a Linear Transformation?
Before we talk about rank and nullity, let’s quickly remember what a linear transformation is. A linear transformation is a rule that maps vectors from one vector space to another (or to itself), while respecting addition and scalar multiplication.
For example, if T is a linear transformation, then:
These rules must hold for all vectors u, v and all scalars c.
Why Do We Care About Rank and Nullity?
Great question! Rank and nullity help us understand how a linear transformation behaves. They tell us:
- How many dimensions of the input space get “collapsed”?
- How big is the output space really?
This isn’t just theory — it’s used in computer graphics, data science, physics, and even engineering! 🤓
Two Special Sets: Kernel and Image
To understand rank and nullity, we first need two important sets related to any linear transformation T: V → W.
1. The Kernel (Null Space)
The kernel of T, written Ker T, is the set of all vectors in V that get mapped to the zero vector in W.
Think of it like a filter: the kernel is what gets filtered out completely.
A) All of ℝ²
B) Only (0, 0)
C) All points where x = y
D) All points where x = −y
Why? Set T(x, y) = (0, 0, 0). Then:
x + y = 0, x − y = 0, y = 0. Solving gives x = 0, y = 0. So only the zero vector maps to zero.
2. The Image (Range)
The image of T, written Im T, is the set of all outputs you can get by applying T to vectors in V.
Both Ker T and Im T are subspaces — they’re closed under addition and scalar multiplication.
What Is Nullity?
Nullity is simply the dimension of the kernel.
If Ker T = {0}, then nullity is 0 — nothing non-zero vanishes.
What Is Rank?
Rank is the dimension of the image.
It tells you how many “independent directions” the transformation can produce in the output.
The Rank-Nullity Theorem – The Heart of This Lesson ❤️
Here’s the magic formula that ties everything together:
Think of it like baking: you start with 100g of flour (input). Some becomes cake (output), some becomes crumbs and waste (kernel). Total = cake + crumbs.
Transformation: T(a, b) = a + b → output in ℝ.
Then:
– Ker T = {(a, −a) | a ∈ ℝ} → dimension 1 (nullity = 1)
– Im T = ℝ → dimension 1 (rank = 1)
– dim(ℝ²) = 2 = 1 + 1 ✅
Deep Walkthrough: Example 5.2.1 from the Textbook
Let’s go step-by-step through the first example in your notes:
Given: G: ℝ³ → ℝ² defined by
\[
G(x, y, z) = (z – y, y + x)
\]
Step 1: Show G is linear
We must check:
- G(u + v) = G(u) + G(v)
- G(cu) = cG(u)
Let u = (x₁, y₁, z₁), v = (x₂, y₂, z₂)
Similarly for scalar multiplication — it works! So G is linear. ✅
Step 2: Find Ker G
We solve G(x, y, z) = (0, 0):
From the first: z = y
From the second: x = −y
So any vector in kernel looks like:
Thus, Ker G is all scalar multiples of (−1, 1, 1). So it’s a line through the origin → dimension = 1.
✅ Nullity = 1
Step 3: Find Im G
We ask: what vectors (a, b) ∈ ℝ² can be written as (z − y, y + x) for some x, y, z?
Notice: we can choose x, y, z freely! For example:
- To get (1, 0): set z − y = 1, y + x = 0 → e.g., y = 0, z = 1, x = 0
- To get (0, 1): set z − y = 0, y + x = 1 → e.g., y = 0, z = 0, x = 1
So both standard basis vectors of ℝ² are in the image → the whole ℝ² is covered!
✅ Im G = ℝ² → dimension = 2 → Rank = 2
Step 4: Check Rank-Nullity Theorem
Input space: ℝ³ → dimension = 3
Rank + Nullity = 2 + 1 = 3 ✅
This confirms the theorem! 😃
Another Example: When Kernel Is Just Zero
Consider T: ℝ² → ℝ³ defined by T(a, b) = (a + b, a − b, b).
Set T(a, b) = (0, 0, 0):
- a + b = 0
- a − b = 0
- b = 0
From 3rd: b = 0 → from 1st: a = 0. So Ker T = {(0, 0)} → nullity = 0.
Image: outputs are of the form (a + b, a − b, b). You can show this spans a 2D plane in ℝ³ → rank = 2.
Check: dim(ℝ²) = 2 = 2 + 0 ✅
A) 0
B) 1
C) 2
D) 3
Solve L(x, y, z) = (0, 0, 0):
– x + y = 0 → x = -y
– 2y = 0 → y = 0 → x = 0
– z is free! So vectors like (0, 0, z) are in kernel.
Thus, Ker L = {(0, 0, z) | z ∈ ℝ} → dimension 1 → nullity = 1.
How Rank and Nullity Tell Us About One-to-One and Onto
Here’s how to use them:
- If nullity = 0, then Ker T = {0} → T is one-to-one.
- If rank = dim(W), then Im T = W → T is onto.
And if both happen and dim(V) = dim(W), then T is invertible! 😮
Practice: Finding Rank and Nullity from a Formula
Let’s try a harder one:
T: ℝ⁴ → ℝ³ defined by:
\[
T(a, b, c, d) = (a – b + c + d,\; a + 2c – d,\; a + b + 3c – 3d)
\]
Find kernel: Set output = (0, 0, 0). Solve the system:
Add (1) and (3):
\[
2a + 4c – 2d = 0 \Rightarrow a + 2c – d = 0
\]
That’s exactly equation (2)! So we only have 2 independent equations.
Let’s use (2): a = d − 2c
Plug into (1): (d − 2c) − b + c + d = 0 → 2d − c − b = 0 → b = 2d − c
So any vector in kernel is:
Two free parameters → kernel is spanned by two vectors → nullity = 2.
Now, dim(ℝ⁴) = 4, so by rank-nullity:
\[
\text{Rank} = 4 – 2 = 2
\]
✅ Done! No need to find the image directly.
Summary of Key Ideas
- Kernel: inputs that map to zero → measure of “loss”.
- Image: actual outputs → measure of “reach”.
- Nullity = dim(kernel), Rank = dim(image).
- Rank-Nullity Theorem: dim(input) = rank + nullity.
- Nullity = 0 ⇨ one-to-one; Rank = dim(output) ⇨ onto.
Final Thought
These ideas are not just for exams — they’re the foundation of modern mathematics and technology. Whether you’re studying AI, physics, or economics, you’ll meet rank and nullity again. So master them now! 💪
Need more help? Check out these free resources:
- Rank–Nullity Theorem – Wikipedia
- Linear Algebra – Khan Academy
- Ethio Temari – Free Ethiopian Educational Content
- Join our community: Telegram, YouTube, TikTok
Keep practicing, stay curious, and never stop asking “why?” — that’s how great mathematicians are made! 😊
Rank and Nullity of a Linear Transformation – Full Exam Quiz
Hello, future mathematicians! 😊 This quiz covers Section 5.2: The Rank and Nullity of a Linear Transformation—a key topic in Linear Algebra I. These ideas connect matrices, vector spaces, and functions in a powerful way.
We’ve designed 20 multiple-choice, 15 matching, 10 true/false, 3 list-and-explain, and 2 fill-in-the-blank questions—all at exam level. Every question includes a “Show Answer” button with a clear explanation. No fluff, just learning!
Need extra help? Visit Ethio Temari, or follow them on Telegram, YouTube, and TikTok. You can also review Wikipedia or Khan Academy.
1. Multiple-Choice Questions (20 Questions)
What is the null space of a linear transformation \( T: V \to W \)?
Answer: B) All vectors in \( V \) that map to zero
The null space (or kernel) is \( \{ v \in V \mid T(v) = 0_W \} \). It lives in the domain \( V \).
What is the rank of a linear transformation \( T \)?
Answer: C) Dimension of the range (image)
Rank = \( \dim(\text{range}(T)) = \dim(\text{im}(T)) \).
If \( T: \mathbb{R}^4 \to \mathbb{R}^3 \) is linear and \( \text{nullity}(T) = 2 \), what is \( \text{rank}(T) \)?
Answer: B) 2
Rank-Nullity Theorem: \( \dim(\text{domain}) = \text{rank}(T) + \text{nullity}(T) \).
So \( 4 = \text{rank}(T) + 2 \Rightarrow \text{rank}(T) = 2 \).
Let \( T: \mathbb{R}^n \to \mathbb{R}^m \) be defined by \( T(x) = Ax \), where \( A \) is a 5 × 7 matrix. What is the domain of \( T \)?
Answer: B) \( \mathbb{R}^7 \)
Matrix \( A \) is 5 × 7 → input vectors must have 7 entries → domain = \( \mathbb{R}^7 \).
If a linear transformation \( T: V \to W \) is one-to-one, what must be true?
Answer: B) \( \text{nullity}(T) = 0 \)
One-to-one ⇔ kernel = {0} ⇔ nullity = 0.
What is the nullity of the zero transformation \( T: \mathbb{R}^3 \to \mathbb{R}^2 \) (where \( T(x) = 0 \) for all \( x \))?
Answer: D) 3
Every vector maps to 0 → null space = entire domain = \( \mathbb{R}^3 \) → nullity = 3.
If \( T: \mathbb{R}^5 \to \mathbb{R}^5 \) and \( \text{rank}(T) = 5 \), which is true?
Answer: C) \( T \) is invertible
Rank = 5 = dim(domain) ⇒ nullity = 0 ⇒ one-to-one.
Rank = 5 = dim(codomain) ⇒ onto.
So \( T \) is bijective → invertible.
Let \( A \) be a 4 × 6 matrix. What is the maximum possible rank of the transformation \( T(x) = Ax \)?
Answer: A) 4
Rank ≤ min(rows, columns) = min(4,6) = 4.
Also, rank = dim(image) ≤ dim(codomain) = 4.
If \( \text{nullity}(T) = \dim(V) \), what is the rank of \( T \)?
Answer: C) 0
Rank + Nullity = dim(V) ⇒ Rank + dim(V) = dim(V) ⇒ Rank = 0.
Let \( T: P_2 \to \mathbb{R}^2 \) be defined by \( T(a + bx + cx^2) = (a, b) \). What is the nullity of \( T \)?
Answer: B) 1
Domain = \( P_2 \): polynomials of degree ≤2 → dim = 3.
\( T(p) = (0,0) \) ⇔ a = 0, b = 0 → p(x) = cx² → null space = span{x²} → dim = 1 → nullity = 1.
Which statement is always true for any linear \( T: V \to W \)?
Answer: C) \( \text{rank}(T) + \text{nullity}(T) = \dim(V) \)
This is the Rank-Nullity Theorem—always true for finite-dimensional V.
If \( T \) is onto, what must be true?
Answer: C) \( \text{rank}(T) = \dim(W) \)
Onto ⇔ range = W ⇔ dim(range) = dim(W) ⇔ rank = dim(W).
Let \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) have matrix \( A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} \). What is nullity(T)?
Answer: B) 1
Solve \( Ax = 0 \): \( x_1 = 0, x_2 = 0, x_3 \) free → null space = span{(0,0,1)} → dim = 1.
If \( \dim(V) = 5 \) and \( T: V \to W \) has rank 3, what is nullity(T)?
Answer: A) 2
Rank + Nullity = dim(V) ⇒ 3 + nullity = 5 ⇒ nullity = 2.
What is the null space of the derivative transformation \( D: P_3 \to P_2 \), where \( D(p) = p’ \)?
Answer: A) All constant polynomials
Derivative is zero ⇔ polynomial is constant → null space = span{1} → dim = 1.
If a linear transformation has a trivial kernel, it is:
Answer: C) One-to-one
Trivial kernel = {0} ⇔ one-to-one.
Let \( T: \mathbb{R}^n \to \mathbb{R}^m \). If \( n > m \), which is always true?
Answer: C) Nullity(T) ≥ 1
Rank ≤ m < n ⇒ Nullity = n − Rank ≥ n − m ≥ 1.
The rank of \( T \) equals the rank of its standard matrix \( A \). Is this true?
Answer: C) Always true
By definition, the rank of \( T \) is dim(im(T)) = dim(col(A)) = rank(A).
If \( T: \mathbb{R}^4 \to \mathbb{R}^4 \) and \( \text{nullity}(T) = 1 \), is \( T \) onto?
Answer: A) Yes
Rank = 4 − 1 = 3? Wait—dim = 4, nullity = 1 → rank = 3.
But codomain is \( \mathbb{R}^4 \), so rank = 3 < 4 → NOT onto. Correction:
Rank = 3 ≠ 4 → not onto. So answer is B) No.
Correct Answer: B) No
Rank = 3 < dim(codomain) = 4 → not onto.
Let \( T: M_{2 \times 2} \to \mathbb{R} \) be \( T(A) = \text{trace}(A) \). What is nullity(T)?
Answer: C) 3
Domain = \( M_{2×2} \): dim = 4.
Range = \( \mathbb{R} \): rank = 1 (since trace can be any real number).
So nullity = 4 − 1 = 3.
2. Matching Questions (15 Questions)
Match each term to its correct definition.
| Term | Definition |
|---|---|
| 1. Rank of T | A. Set of all v such that T(v) = 0 |
| 2. Null space | B. Dimension of the range of T |
| 3. Nullity | C. Dimension of the null space |
| 4. One-to-one | D. T(u) = T(v) implies u = v |
| 5. Onto | E. Range(T) = codomain |
| 6. Kernel | F. Same as null space |
| 7. Image | G. Set of all T(v) for v in V |
| 8. Rank-Nullity Theorem | H. dim(V) = rank(T) + nullity(T) |
| 9. Trivial kernel | I. Kernel = {0} |
| 10. Standard matrix | J. Matrix A such that T(x) = Ax |
| 11. Domain | K. Space where input vectors live |
| 12. Codomain | L. Space where output vectors live |
| 13. Range | M. Same as image |
| 14. Linear transformation | N. Function preserving addition and scalar multiplication |
| 15. Zero transformation | O. Sends every vector to zero |
Correct Pairings:
1–B, 2–A, 3–C, 4–D, 5–E, 6–F, 7–G, 8–H, 9–I, 10–J, 11–K, 12–L, 13–M, 14–N, 15–O.
Explanation: These are core definitions from Section 5.2. Remember: kernel = null space, image = range, and Rank-Nullity is your best friend!
3. True/False Questions (10 Questions)
The nullity of a linear transformation can be negative.
Answer: False
Nullity = dimension of a subspace → always ≥ 0.
If rank(T) = 0, then T is the zero transformation.
Answer: True
Rank = 0 ⇒ image = {0} ⇒ T(v) = 0 for all v.
Every one-to-one transformation is onto.
Answer: False
Only true if domain and codomain have the same finite dimension. Counterexample: \( T: \mathbb{R}^2 \to \mathbb{R}^3 \), \( T(x,y) = (x,y,0) \) is one-to-one but not onto.
If T is onto, then nullity(T) = 0.
Answer: False
Onto relates to rank, not nullity. Example: \( T: \mathbb{R}^3 \to \mathbb{R}^2 \), projection → onto but nullity = 1.
The Rank-Nullity Theorem applies only to transformations from \( \mathbb{R}^n \) to \( \mathbb{R}^m \).
Answer: False
It applies to any linear transformation between finite-dimensional vector spaces.
If dim(V) = dim(W) and T is one-to-one, then T is onto.
Answer: True
One-to-one ⇒ nullity = 0 ⇒ rank = dim(V) = dim(W) ⇒ onto.
The null space of T is a subspace of W.
Answer: False
Null space is in the domain V, not codomain W.
Rank(T) ≤ dim(V) and Rank(T) ≤ dim(W).
Answer: True
Image is a subspace of W → dim(image) ≤ dim(W).
Also, rank = dim(V) − nullity ≤ dim(V).
If T is invertible, then nullity(T) = 0.
Answer: True
Invertible ⇒ one-to-one ⇒ kernel = {0} ⇒ nullity = 0.
The derivative map \( D: P_n \to P_{n-1} \) has nullity 1.
Answer: True
Kernel = constant polynomials → dimension = 1 → nullity = 1.
4. List-and-Explain Questions (3 Questions)
List the three key properties that define a linear transformation \( T: V \to W \). Explain each with a short example.
Answer:
A function \( T: V \to W \) is linear if:
- Additivity: \( T(u + v) = T(u) + T(v) \) for all \( u, v \in V \).
Example: \( T(x,y) = (x,0) \) → \( T((1,1)+(0,1)) = T(1,2) = (1,0) = (1,0)+(0,0) = T(1,1)+T(0,1) \). - Homogeneity: \( T(cu) = cT(u) \) for all scalars \( c \) and \( u \in V \).
Example: Same \( T \), \( T(2·(1,1)) = T(2,2) = (2,0) = 2·(1,0) = 2T(1,1) \). - Preserves zero: \( T(0_V) = 0_W \).
This follows from homogeneity: \( T(0) = T(0·v) = 0·T(v) = 0 \).
If any one fails, \( T \) is not linear.
State the Rank-Nullity Theorem. Then use it to find the nullity of a transformation \( T: \mathbb{R}^6 \to \mathbb{R}^4 \) with rank 3.
Answer:
Rank-Nullity Theorem: If \( T: V \to W \) is linear and \( V \) is finite-dimensional, then
\[
\dim(V) = \text{rank}(T) + \text{nullity}(T)
\]
Here, \( \dim(V) = 6 \), \( \text{rank}(T) = 3 \).
So \( 6 = 3 + \text{nullity}(T) \Rightarrow \text{nullity}(T) = 3 \).
Let \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) be defined by \( T(x, y, z) = (x + y, y + z, x + z) \).
(a) Find the standard matrix of \( T \).
(b) Find rank and nullity of \( T \).
(c) Is \( T \) one-to-one? Onto?
Answer:
(a) Apply \( T \) to standard basis:
\( T(1,0,0) = (1,0,1) \)
\( T(0,1,0) = (1,1,0) \)
\( T(0,0,1) = (0,1,1) \)
So standard matrix:
\[
A = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix}
\]
(b) Row reduce \( A \):
\( R_3 \to R_3 – R_1 \): \( \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & -1 & 1 \end{bmatrix} \)
\( R_3 \to R_3 + R_2 \): \( \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 2 \end{bmatrix} \)
3 pivots → rank = 3 → nullity = 3 − 3 = 0.
(c) Nullity = 0 ⇒ one-to-one.
Rank = 3 = dim(codomain) ⇒ onto.
So \( T \) is both one-to-one and onto (invertible).
5. Fill-in-the-Blank Questions (2 Questions)
The dimension of the null space of a linear transformation is called its ________.
Answer: nullity
This is the standard term used in linear algebra.
For any linear transformation \( T: V \to W \), the sum of the rank and nullity equals ________.
Answer: the dimension of V
This is the statement of the Rank-Nullity Theorem.
Great job! 🎯 You’ve just practiced one of the most important theorems in linear algebra. Keep reviewing with Ethio Temari and other free resources. Your understanding of rank and nullity will help you in differential equations, data science, and beyond!