Matrix rank and null space in MATLAB - Step-by-Step Execution

Concept Flow - Matrix rank and null space

Start with matrix A

↓

Calculate rank(A)

↓

Calculate null space of A

↓

Interpret results

↓

End

We start with a matrix A, find its rank, then find its null space, and finally interpret what these results mean.

Execution Sample

MATLAB

A = [1 2 3; 4 5 6; 7 8 9];
r = rank(A);
N = null(A);

This code finds the rank and null space of matrix A.

Execution Table

Step	Action	Value/Result	Explanation
1	Define matrix A	[[1 2 3]; [4 5 6]; [7 8 9]]	Matrix A is set with 3 rows and 3 columns
2	Calculate rank(A)	2	Rank is 2 because rows are linearly dependent
3	Calculate null(A)	[[-1; 2; -1]]	Null space is a vector that satisfies A*x=0
4	Interpret rank	2	Rank 2 means 2 independent rows/columns
5	Interpret null space	1-dimensional	Null space dimension = number of columns - rank = 1
6	End	-	Finished calculations

💡 Finished after calculating rank and null space of matrix A

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	Final
A	undefined	[[1 2 3]; [4 5 6]; [7 8 9]]	[[1 2 3]; [4 5 6]; [7 8 9]]	[[1 2 3]; [4 5 6]; [7 8 9]]	[[1 2 3]; [4 5 6]; [7 8 9]]
r	undefined	undefined	2	2	2
N	undefined	undefined	undefined	[[-1]; [2]; [-1]]	[[-1]; [2]; [-1]]

Key Moments - 3 Insights

Why is the rank of matrix A equal to 2 and not 3?

Because the rows of A are linearly dependent, as shown in step 2 of the execution_table where rank(A) is 2, meaning only 2 rows add new information.

What does the null space vector represent in this context?

The null space vector shown in step 3 is a vector x where A*x = 0, meaning it lies in the space where the matrix collapses to zero, indicating dependencies.

How is the dimension of the null space related to the rank?

The null space dimension equals the number of columns minus the rank, as explained in step 5, so here 3 - 2 = 1 dimension.

Visual Quiz - 3 Questions

Look at the execution_table, what is the rank of matrix A at step 2?

A 1

B 3

C 2

D 0

💡 Hint

Check the 'Value/Result' column at step 2 in the execution_table.

At which step do we find the null space vector of matrix A?

A Step 1

B Step 3

C Step 2

D Step 4

💡 Hint

Look for the step where 'Calculate null(A)' is performed in the execution_table.

If matrix A had full rank 3, what would be the dimension of the null space?

A 0

B 3

C 1

D 2

💡 Hint

Recall from key_moments that null space dimension = columns - rank.

Concept Snapshot

Matrix rank: number of independent rows/columns.
Null space: vectors x where A*x=0.
rank(A) + nullity(A) = number of columns.
Use rank(A) and null(A) in MATLAB.
Rank shows dimension of image space.
Null space shows dependencies.

Full Transcript

We start with a matrix A defined as a 3x3 matrix. Then we calculate its rank using MATLAB's rank function, which returns 2 because the rows are not all independent. Next, we calculate the null space using null(A), which returns a vector that satisfies A*x=0. The rank tells us how many independent rows or columns the matrix has, and the null space dimension is the difference between the number of columns and the rank. This helps us understand the structure of the matrix and its linear dependencies.

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