Linear systems of equations, represented as Ax = b, are a fundamental concept in mathematics and have wide-ranging applications in various fields. In this article, we will explore how the singular value decomposition (SVD) can be used to solve such systems, even when the matrix A is not square. Specifically, we will delve into the conditions under which a linear system either has no solutions, a unique solution, or infinitely many solutions.
Contents
The Column Space of A
To determine the existence of a solution to Ax = b, we must analyze whether vector b lies within the column space of matrix A. The column space, denoted by “column of A,” represents all possible linear combinations of the column vectors of A. In other words, if we can find a vector x that satisfies Ax = b, it implies that b resides within the span of the column vectors of A.
However, there are cases where b is not within the column space of A. For example, consider a degenerate A matrix with only one direction in its column space. In such situations, no solution exists for Ax = b.
Overdetermined and Underdetermined Systems
In an overdetermined system, where the number of equations exceeds the number of unknowns, one might not expect an exact solution for Ax = b. However, if b is one of the columns of A, thereby lying within the column space of A, a solution does exist.
Conversely, in an underdetermined system, where the number of unknowns surpasses the number of equations, multiple solutions are possible. This arises when the A matrix has a large null space, and b lies within the column space of A. Consequently, the system has infinitely many solutions.
The Fundamental Subspaces of A
To gain deeper insights into these scenarios, we need to explore three additional subspaces associated with matrix A: the orthogonal complement, row space, and null space.
The orthogonal complement represents the set of vectors orthogonal to the span of A. More specifically, the orthogonal complement is the kernel of A transpose. If b lies within the orthogonal complement, no solution exists for Ax = b.
The row space of A comprises all possible linear combinations of its row vectors. It is equivalent to the column space of the transpose of A.
The null space, denoted as the kernel of A, consists of all vectors x that result in Ax = 0. It is a crucial subspace that helps determine whether a linear system has no solutions or infinitely many solutions.
Summary of Solution Possibilities
Based on the concepts discussed above, we can summarize the solution possibilities for a linear system Ax = b:
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No solutions: If b is not within the column space of A or lies within the orthogonal complement, no solution exists.
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Unique solution: A unique solution exists when b resides within the column space of A, and the dimension of the null space of A is 0.
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Infinitely many solutions: If b lies within the column space of A, and the dimension of the null space of A is not 0, infinitely many solutions exist.
It’s important to note that while these general rules hold true, specific cases may deviate from them. Nonetheless, understanding the column space, orthogonal complement, row space, and null space of A can provide valuable insights into the behavior of linear systems of equations.
FAQs
Q: What is the column space of A?
A: The column space of A represents all possible linear combinations of the column vectors of matrix A.
Q: What is the null space of A?
A: The null space of A, denoted as the kernel of A, consists of all vectors x that result in Ax = 0.
Q: When does a linear system have no solutions?
A: A linear system has no solutions if b is not within the column space of A or lies within the orthogonal complement.
Q: When does a linear system have a unique solution?
A: A linear system has a unique solution when b resides within the column space of A, and the dimension of the null space of A is 0.
Q: When does a linear system have infinitely many solutions?
A: A linear system has infinitely many solutions if b lies within the column space of A, and the dimension of the null space of A is not 0.
Conclusion
Understanding linear systems of equations is fundamental to the field of mathematics and has broad applications in various domains. By analyzing the column space, orthogonal complement, row space, and null space of matrix A, we can determine the existence and nature of solutions for a given linear system. This knowledge serves as a building block for more advanced concepts, such as linear regression in machine learning. To explore these topics further, visit the Techal website.
