Matrix rank and null space in MATLAB - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the rank of a matrix?

The rank of a matrix is the number of linearly independent rows or columns it has. It tells us how many dimensions the matrix's data spans.

beginner

What does the null space of a matrix represent?

The null space is the set of all vectors that, when multiplied by the matrix, give the zero vector. It shows directions where the matrix 'flattens' vectors to zero.

beginner

How do you find the rank of a matrix in MATLAB?

Use the command rank(A), where A is your matrix. It returns the number of independent rows or columns.

beginner

How do you find the null space of a matrix in MATLAB?

Use the command null(A). It returns a matrix whose columns form a basis for the null space of A.

intermediate

If a matrix has full rank, what can you say about its null space?

If a matrix has full rank, its null space contains only the zero vector. This means no non-zero vector maps to zero.

What MATLAB command gives the rank of matrix A?

A rank(A)

B null(A)

C det(A)

D inv(A)

The rank(A) command returns the number of linearly independent rows or columns of matrix A.

What does the null space of a matrix contain?

A All eigenvectors of the matrix

B All vectors mapped to zero by the matrix

C All vectors orthogonal to the matrix

D All vectors with positive entries

The null space contains all vectors that the matrix sends to the zero vector.

If rank(A) = n for an n×n matrix, what is the dimension of the null space?

A 1

B n

C 0

D n-1

Full rank means the null space only has the zero vector, so its dimension is zero.

Which MATLAB command returns a basis for the null space of A?

A eig(A)

B rank(A)

C svd(A)

D null(A)

The null(A) command returns vectors that form a basis for the null space.

What does a rank less than the number of columns imply about the null space?

A Null space has non-zero vectors

B Null space is empty

C Matrix is invertible

D Matrix is diagonal

If rank is less than columns, there are non-zero vectors in the null space.

Explain in your own words what the rank of a matrix tells us and how to find it in MATLAB.

Think about how many directions the matrix 'covers' without overlap.

Describe the null space of a matrix and how MATLAB helps you find it.

Consider what vectors disappear when multiplied by the 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