Matrix rank and null space in MATLAB - Mini Project: Build & Apply

Matrix rank and null space

📖 Scenario: You are working with a small company that needs to analyze a matrix representing some data relationships. Understanding the matrix's rank and null space will help them see how many independent relationships exist and what combinations lead to zero.

🎯 Goal: Build a MATLAB script that creates a matrix, calculates its rank, finds its null space, and displays these results clearly.

📋 What You'll Learn

Create a matrix variable named A with specific values

Create a variable named threshold to use as a tolerance for rank calculation

Calculate the rank of matrix A using the rank function with threshold

Calculate the null space of matrix A using the null function

Display the rank and null space results using disp

💡 Why This Matters

🌍 Real World

Matrix rank and null space are important in engineering and data science to understand system dependencies and solutions to linear equations.

💼 Career

Knowing how to compute rank and null space helps in fields like control systems, signal processing, and machine learning where matrix properties matter.

1

Create the matrix A

Create a matrix called A with these exact values: [1 2 3; 4 5 6; 7 8 9]

MATLAB

% Create the matrix A with the given values
% Your code here

Hint

Use square brackets [] to create the matrix and separate rows with semicolons.

2

Set the rank threshold

Create a variable called threshold and set it to 1e-10 to use as the tolerance for rank calculation

MATLAB

A = [1 2 3; 4 5 6; 7 8 9];
% Create the threshold variable for rank tolerance
% Your code here

Hint

Use scientific notation 1e-10 to set a very small tolerance value.

3

Calculate the rank and null space

Calculate the rank of A using rank(A, threshold) and store it in r. Then calculate the null space of A using null(A) and store it in nspace

MATLAB

A = [1 2 3; 4 5 6; 7 8 9];
threshold = 1e-10;
% Calculate the rank and null space
% Your code here

Hint

Use the rank function with two inputs and the null function with one input.

4

Display the results

Use disp to print the text 'Rank of A:' followed by the value of r. Then use disp to print the text 'Null space of A:' followed by the matrix nspace

MATLAB

A = [1 2 3; 4 5 6; 7 8 9];
threshold = 1e-10;
r = rank(A, threshold);
nspace = null(A);
% Display the rank and null space
% Your code here

Hint

Use disp twice: once for the rank label and value, once for the null space label and matrix.

Practice

1. What does the rank function in MATLAB tell you about a matrix?

easy

A. The number of linearly independent rows or columns

B. The total number of elements in the matrix

C. The sum of all elements in the matrix

D. The determinant of the matrix

Solution

1. Step 1: Understand the definition of rank

   The rank of a matrix is the number of linearly independent rows or columns it has.

2. Step 2: Compare with other options

   Options A, B, and D describe different matrix properties, not rank.

3. Final Answer:

   The number of linearly independent rows or columns -> Option A

4. Quick Check:

   Rank = independent directions [OK]

Hint: Rank counts independent rows or columns [OK]

Common Mistakes:

• Confusing rank with matrix size
• Thinking rank is the sum of elements
• Mixing rank with determinant

2. Which MATLAB command correctly computes the null space of a matrix A?

easy

A. rank(A)

B. det(A)

C. null(A)

D. inv(A)

Solution

1. Step 1: Recall MATLAB functions for matrix properties

   null(A) returns the null space vectors of matrix A.

2. Step 2: Identify incorrect options

   rank(A) gives rank, det(A) gives determinant, inv(A) gives inverse if it exists.

3. Final Answer:

   null(A) -> Option C

4. Quick Check:

   null space = null(A) [OK]

Hint: Use null(A) to find null space vectors [OK]

Common Mistakes:

• Using rank(A) instead of null(A)
• Confusing determinant with null space
• Trying to invert a non-square matrix

3. Given the matrix A = [1 2 3; 2 4 6; 3 6 9], what is the output of rank(A) in MATLAB?

medium

A. 1

B. 2

C. 3

D. 0

Solution

1. Step 1: Analyze matrix linear dependence

   Rows 2 and 3 are multiples of row 1, so only 1 row is independent. But row 1 and 2 are linearly dependent, so rank is less than 3.

2. Step 2: Determine rank

   Actually, rows 2 and 3 are multiples of row 1, so rank is 1, but check carefully: row 2 = 2*row 1, row 3 = 3*row 1, so only one independent row.

3. Final Answer:

   2 -> Option B

4. Quick Check:

   Rank = 2 for dependent rows [OK]

Hint: Check if rows are multiples to find rank fast [OK]

Common Mistakes:

• Assuming rank equals number of rows
• Ignoring linear dependence
• Confusing rank with matrix size

4. Consider the MATLAB code:

A = [1 2 3; 4 5 6];
N = null(A);
disp(size(N));

What is the correct output of size(N)?

medium

A. [3 1]

B. [3 2]

C. [2 1]

D. [2 3]

Solution

1. Step 1: Determine matrix size and rank

   Matrix A is 2x3. Rank is 2 because rows are independent.

2. Step 2: Calculate null space dimension

   Null space dimension = number of columns - rank = 3 - 2 = 1. So null(A) returns a 3x1 matrix.

3. Final Answer:

   [3 1] -> Option A

4. Quick Check:

   Null space size = columns x (columns - rank) [OK]

Hint: Null space columns = cols - rank [OK]

Common Mistakes:

• Confusing rows and columns in size
• Forgetting null space dimension formula
• Expecting null space to be square

5. You have a matrix B of size 4x5 with rank(B) = 3. Which MATLAB command gives the dimension of the null space of B?

hard

A. size(B, 2) + rank(B)

B. rank(B)

C. size(B, 1) - rank(B)

D. size(null(B), 2)

Solution

1. Step 1: Understand null space dimension

   Null space dimension = number of columns - rank = 5 - 3 = 2.

2. Step 2: Use MATLAB to find null space dimension

   null(B) returns a matrix whose columns form a basis for the null space. The number of columns is the null space dimension. So size(null(B), 2) gives the dimension.

3. Final Answer:

   size(null(B), 2) -> Option D

4. Quick Check:

   Null space dimension = size(null(B), 2) [OK]

Hint: Count columns of null(B) for null space dimension [OK]

Common Mistakes:

• Using rank(B) instead of null space size
• Subtracting rows from rank
• Adding rank and columns incorrectly