Understanding the Orthogonality between Row Space and Null Space of a Matrix

Introduction

In linear algebra, the concepts of row space and null space are fundamental. They provide important insights into the structure and properties of a matrix. This article will explore the relationship between the row space and null space, focusing specifically on their orthogonality. Understanding this relationship is crucial for various applications in mathematics, engineering, and data science.

The Matrix and Its Spaces

Consider a matrix ( A ). The row space of ( A ) is the span of its row vectors, while the null space of ( A ), denoted by ( N(A) ), is the set of all vectors ( x ) such that ( Ax 0 ).

Row Space and Null Space Orthogonality

The key concept here is that the row space of a matrix ( A ) and the null space of the same matrix ( A ) are orthogonal to each other. This relationship can be proven through a series of logical steps and vector properties.

Proof of Orthogonality: Row Space Orthogonal to Null Space

First, let's assume a vector ( x ) is in the null space of ( A ). By definition, this means ( Ax 0 ). Now, let's consider the dot product between ( x ) and each of the rows of ( A ).

Let ( A ) be an ( m times n ) matrix with rows ( r_1, r_2, ldots, r_m ). If ( Ax 0 ), then

[ r_i cdot x 0 quad text{for all } i 1, 2, ldots, m]

This implies that ( x ) is orthogonal to each row vector of ( A ). By definition of orthogonality, a vector is orthogonal to a set of vectors if it is perpendicular to all the vectors in that set. Therefore, ( x ) is orthogonal to the row space of ( A ).

Next, we consider any linear combination of the rows of ( A ). If ( x ) is orthogonal to all rows of ( A ), it must also be orthogonal to any linear combination of these rows. Let ( s ) be a linear combination of the rows of ( A ). Therefore,

[ s k_1 r_1 k_2 r_2 cdots k_m r_m]

Given that ( x ) is orthogonal to each ( r_i ), it follows that

[ x cdot s x cdot (k_1 r_1 k_2 r_2 cdots k_m r_m) k_1 (x cdot r_1) k_2 (x cdot r_2) cdots k_m (x cdot r_m) 0]

Thus, ( x ) is orthogonal to ( s ). Since ( s ) is any linear combination of the row vectors of ( A ), we conclude that ( x ) is in the null space and is orthogonal to the row space of ( A ).

Conversely, Null Space Orthogonal to Row Space

Now, let's prove the converse: if a vector ( s ) is perpendicular to the row space of ( A ), then it must be in the null space of ( A ).

If ( s ) is orthogonal to the row space of ( A ), then ( s ) is orthogonal to each of the rows of ( A ). Consider the matrix equation ( As ). Each component of ( As ) can be expressed as the dot product of ( s ) and a row of ( A ). Since ( s ) is orthogonal to each row, each component of ( As ) is zero:

[ (As) cdot r_i s cdot (A cdot r_i) s cdot 0 0 quad text{for all } i 1, 2, ldots, m]

Since ( As ) is the zero vector, we have ( As 0 ), which means ( s ) is in the null space of ( A ).

Conclusion

Thus, we have established that the row space and null space of a matrix ( A ) are orthogonal to each other. This relationship is a fundamental property in linear algebra and has practical applications in many fields, including data analysis and machine learning.

Related Keywords

orthogonality row space null space

About the Author

Alibaba Cloud is renowned for its cutting-edge technologies and innovative solutions in the field of artificial intelligence and cloud computing. This article is authored by Qwen, a language model created by Alibaba Cloud, providing valuable insights into the field of mathematics and its applications.