Find Left Null Space Of Matrix Calculator

What is the Left Null Space of a Matrix?

The left null space of a matrix A, denoted as N(A^T), is the set of all row vectors y^T such that y^TA = 0^T (the zero row vector). Equivalently, it is the null space of the transpose of A, A^T. That is, it’s the set of all column vectors y such that A^Ty = 0 (the zero column vector).

If A is an m x n matrix, its transpose A^T is an n x m matrix. The left null space of A is a subspace of R^m (the space of m-dimensional row vectors, or n-dimensional column vectors if considering A^Ty=0). The dimension of the left null space of A is given by m – r, where r is the rank of A (which is also the rank of A^T).

The left null space is one of the four fundamental subspaces associated with a matrix A, the others being the column space (C(A)), the null space (N(A)), and the row space (C(A^T)). The row space and null space are orthogonal complements in Rⁿ, while the column space and left null space are orthogonal complements in R^m.

Who should use it?

Students of linear algebra, engineers, scientists, and anyone working with systems of linear equations, vector spaces, or data analysis involving matrices often need to understand and find the left null space of a matrix. It’s crucial for understanding the relationships between the rows of a matrix and the solutions to linear systems.

Common Misconceptions

A common misconception is confusing the left null space (N(A^T)) with the (right) null space (N(A)). The null space N(A) consists of vectors x such that Ax = 0, while the left null space N(A^T) consists of vectors y such that A^Ty = 0 (or y^TA = 0^T).

Left Null Space of a Matrix Formula and Mathematical Explanation

To find the left null space of a matrix A (m x n), we follow these steps:

1. Transpose the Matrix A: Find A^T, which will be an n x m matrix.
2. Set up the Homogeneous System: We want to solve A^Ty = 0, where y is a column vector with m components.
3. Row Reduce A^T: Perform Gaussian elimination on A^T to transform it into its Reduced Row Echelon Form (RREF).
4. Identify Pivot and Free Variables: In the RREF of A^T, columns with leading 1s correspond to pivot variables, and columns without leading 1s correspond to free variables among the components of y.
5. Express Pivot Variables: Write the pivot variables in terms of the free variables from the equations represented by the RREF of A^T.
6. Form Basis Vectors: Construct the basis vectors for the null space of A^T by setting each free variable to 1 (and others to 0) one at a time and finding the corresponding values of the pivot variables. These vectors span the left null space of matrix A.

The dimension of the left null space is equal to the number of free variables in the system A^Ty = 0, which is m – rank(A^T) = m – rank(A).

Variables Table

Variable	Meaning	Unit	Typical range
A	The original m x n matrix	Matrix	Real numbers
A^T	The transpose of A (n x m matrix)	Matrix	Real numbers
m	Number of rows of A	Integer	≥ 1
n	Number of columns of A	Integer	≥ 1
r	Rank of A (and A^T)	Integer	0 ≤ r ≤ min(m, n)
dim(N(A^T))	Dimension of the left null space	Integer	m – r
y	Vector in the left null space (m x 1)	Vector	Real numbers

The table above summarizes the key elements when working with the left null space of a matrix.

Practical Examples (Real-World Use Cases)

Example 1: Finding the Left Null Space

Let’s consider matrix A:


A = [[1, 2, 3],
[4, 5, 6]]


Here, m=2, n=3. A is 2×3.

1. Transpose A:


AT = [[1, 4],
[2, 5],
[3, 6]]


A^T is 3×2.

2. Solve A^Ty = 0. We row reduce A^T:


[[1, 4], R2 = R2 - 2R1 [[1, 4], R3 = R3 - 3R1 [[1, 4], R2 = -1/3 * R2 [[1, 4], R1=R1-4R2 [[1, 0],
[2, 5], ---------> [0,-3], ---------> [0,-3], ----------> [0, 1], --------> [0, 1],
[3, 6]] [3, 6]] [0,-6]] [0,-6]] R3=R3+6R2 [0, 0]]


RREF(A^T) is [[1, 0], [0, 1], [0, 0]]. Both columns are pivot columns. There are no free variables. The only solution to A^Ty = 0 is y = [0, 0]^T. So, the left null space of A is just the zero vector {0}, and its dimension is 0. (m-r = 2-2=0)

Example 2: A Matrix with a Non-trivial Left Null Space

Let’s consider matrix A:


A = [[1, 0, 1],
[2, 0, 2]]


Here m=2, n=3.

1. Transpose A:


AT = [[1, 2],
[0, 0],
[1, 2]]


A^T is 3×2.

2. Solve A^Ty = 0 for y=[y1, y2, y3]^T, which isn’t right. y is m x 1, so y=[y1, y2]^T. A^T is 3×2, y is 2×1. So we solve A^Ty = 0 where y = [y1, y2]^T. Wait, A is m x n, A^T is n x m (3×2), y must be m x 1 (2×1).
Let’s recheck the definition: y^TA = 0^T, y^T is 1xm, so y is mx1. So y = [y1, y2]^T.
A^Ty = 0 where A^T is 3×2 and y is 2×1.
[[1, 2], [0, 0], [1, 2]] [y1, y2]^T = [0,0,0]^T
y1 + 2y2 = 0
0 = 0
y1 + 2y2 = 0
RREF of A^T: [[1, 2], [0, 0], [0, 0]]. Rank = 1.
y1 = -2y2. y2 is free. Let y2=t, y1=-2t. y = [-2t, t]^T = t[-2, 1]^T.
The left null space is spanned by [-2, 1]^T or row vector [-2, 1]. Dimension = m-r = 2-1 = 1.

Let’s take a 3×3 matrix A to have a non-trivial left null space more clearly.


A = [[1, 2, 3],
[0, 1, 1],
[1, 3, 4]]

m=3, n=3.
A^T = [[1, 0, 1], [2, 1, 3], [3, 1, 4]]
Row reduce A^T:
[[1, 0, 1], R2=R2-2R1 [[1, 0, 1], R3=R3-3R1 [[1, 0, 1], R3=R3-R2 [[1, 0, 1],
[2, 1, 3], ——–> [0, 1, 1], ——–> [0, 1, 1], ——–> [0, 1, 1],
[3, 1, 4]] [0, 1, 1]] [0, 1, 1]] [0, 0, 0]]
RREF(A^T) = [[1, 0, 1], [0, 1, 1], [0, 0, 0]]. Rank=2. Dimension of left null space = m-r = 3-2 = 1.
A^Ty = 0 gives: y1 + y3 = 0, y2 + y3 = 0. y3 is free.
y1 = -y3, y2 = -y3. Let y3=t. y = [-t, -t, t]^T = t[-1, -1, 1]^T.
Basis for left null space: {[-1, -1, 1]^T}.

How to Use This Left Null Space of a Matrix Calculator

1. Enter Matrix Dimensions: Select the number of rows (m) and columns (n) for your matrix A using the dropdowns. The calculator supports matrices up to 4×4.
2. Input Matrix Elements: Enter the numerical values for each element of matrix A into the generated input fields. Ensure you enter valid numbers.
3. Calculate: Click the “Calculate Left Null Space” button.
4. View Results: The calculator will display:
   • The original matrix A.
   • The transpose A^T.
   • The Reduced Row Echelon Form (RREF) of A^T.
   • The rank of A^T.
   • The basis vectors for the left null space of matrix A.
   • The dimension of the left null space (m – rank(A^T)).
   • A chart visualizing the dimensions, rank, and nullity.
5. Interpret Basis: The basis vectors form a set of linearly independent vectors that span the left null space. Any vector in the left null space can be written as a linear combination of these basis vectors.
6. Reset: Click “Reset” to clear the inputs and results and start with default values.
7. Copy: Click “Copy Results” to copy the main results and intermediate values to your clipboard.

Understanding the dimension of the left null space of a matrix is key; if it’s 0, only the zero vector is in it.

Key Factors That Affect Left Null Space Results

• Matrix Dimensions (m and n): The number of rows (m) of A determines the dimension of the space R^m where the left null space resides and influences its maximum possible dimension.
• Rank of the Matrix (r): The rank of A (and A^T) directly determines the dimension of the left null space (m-r). A higher rank means a smaller left null space dimension.
• Linear Independence of Rows of A: The rows of A form the columns of A^T. If the rows of A are linearly independent (when m ≤ n), the rank of A^T is m, and the left null space is trivial ({0}). If rows are dependent, the rank is less than m, and the left null space is non-trivial.
• Values of Matrix Elements: The specific numbers in the matrix determine the relationships between the rows and thus the structure and basis of the left null space of the matrix after row reduction.
• Number of Free Variables: When solving A^Ty = 0, the number of free variables after row reduction equals the dimension of the left null space.
• Pivots in RREF(A^T): The positions of the pivot elements in the RREF of A^T identify the pivot variables, which are expressed in terms of the free variables to find the basis vectors.

Frequently Asked Questions (FAQ)

What is the left null space of a matrix A?

It’s the set of all row vectors y^T such that y^TA = 0^T, or equivalently, the null space of A^T, containing all column vectors y such that A^Ty = 0.

How is the left null space different from the null space?

The left null space N(A^T) involves A^T (or multiplying A from the left by a row vector), while the null space N(A) involves A and multiplying A from the right by a column vector (Ax=0).

What is the dimension of the left null space?

If A is m x n, the dimension of its left null space is m – r, where r is the rank of A.

Why is it called the ‘left’ null space?

Because it involves vectors that multiply A from the left (y^TA = 0^T). It’s associated with the rows of A.

What does it mean if the left null space only contains the zero vector?

It means the rows of matrix A are linearly independent (if m ≤ n and rank=m), and the only linear combination of rows that gives the zero row is the trivial one (all coefficients zero).

How is the left null space related to the column space?

The left null space of A is the orthogonal complement of the column space of A in R^m.

Can any matrix have a non-trivial left null space?

Yes, if the rows of the matrix A are linearly dependent (rank(A) < m), then the left null space will have a dimension greater than zero and be non-trivial.

How do I find a basis for the left null space?

Find the RREF of A^T, solve A^Ty=0, identify free variables, and express the solution in terms of these free variables to get the basis vectors using our left null space of matrix calculator.

Related Tools and Internal Resources

• Null Space Calculator: Find the null space N(A) of a matrix A.
• Matrix Transpose Calculator: Calculate the transpose of a matrix.
• RREF Calculator: Find the Reduced Row Echelon Form of a matrix.
• Matrix Rank Calculator: Determine the rank of a matrix.
• Column Space Calculator: Find the basis for the column space of a matrix.
• Row Space Calculator: Find the basis for the row space of a matrix.

These tools, including the find left null space of matrix calculator, are valuable for understanding linear algebra concepts.