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.