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.