Question

Let \(A_{1}, A_{2}, \ldots, A_{m}\) denote \(n \times n\) matrices. If \(\mathbf{0} \neq \mathbf{y} \in \mathbb{R}^{n}\) and \(A_{1} \mathbf{y}=A_{2} \mathbf{y}=\cdots=A_{m} \mathbf{y}=\mathbf{0},\) show that \(\left\\{A_{1}, A_{2}, \ldots, A_{m}\right\\}\) cannot \(\operatorname{span} \mathbf{M}_{n n}\).

Short answer

The matrices cannot span \(\mathbf{M}_{nn}\) because they cannot form transformations that map \(\mathbf{y}\) to any non-zero vector.

Step-by-step solution

Understand the Problem Statement

We are given a set of matrices \(A_1, A_2, \ldots, A_m\), each of size \(n \times n\), and a non-zero vector \(\mathbf{y}\) such that \(A_i \mathbf{y} = \mathbf{0}\) for all matrices \(i\). We need to show that these matrices cannot span the entire space of all \(n \times n\) matrices, denoted as \(\mathbf{M}_{nn}\).

Recall Definitions

The statement that a set of matrices \(\{A_1, A_2, \ldots, A_m\}\) spans \(\mathbf{M}_{nn}\) means that any \(n \times n\) matrix \(B\) can be expressed as a linear combination of the matrices \(A_i\). This implies that if they span \(\mathbf{M}_{nn}\), they can create any linear transformation from \(\mathbb{R}^n\) to \(\mathbb{R}^n\).

Consider Properties of the Given Matrices

We are told \(A_i \mathbf{y} = \mathbf{0}\) for all \(i\). This means that \(\mathbf{y}\) is in the null space of each matrix \(A_i\). If the matrices spanned all transformations, it would imply an ability to represent any transformation including those not annihilating \(\mathbf{y}\).

Argument by Contradiction

Assume that \(\{A_1, A_2, \ldots, A_m\}\) spans \(\mathbf{M}_{nn}\). Then there must be a matrix \(B\) in the span such that \(B\mathbf{y}\) can be any vector in \(\mathbb{R}^n\). However, since \(A_i \mathbf{y} = \mathbf{0}\) for all \(A_i\), any linear combination of them applied to \(\mathbf{y}\) is also zero. Thus, we cannot form every \(n \times n\) matrix, leading to a contradiction.

Conclusion

Since \(B\mathbf{y} = \mathbf{0}\) for every matrix \(B\) that is a linear combination of \(A_1, A_2, \ldots, A_m\), the assumption that these matrices span \(\mathbf{M}_{nn}\) is incorrect. Thus, the set \(\{A_1, A_2, \ldots, A_m\}\) cannot span \(\mathbf{M}_{nn}\).

Key concepts

Matrix Theory

Matrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. Matrix theory is an essential aspect of linear algebra dealing with various operations and applications of matrices. In this context, the set of all matrices with a given size (such as \(n \times n\)) can perform numerous transformations.
Understanding matrices involves recognizing their role in solving systems of linear equations, performing linear transformations, and representing data.
Here are some key aspects of matrix theory related to this exercise:

• Matrix Addition and Scalar Multiplication: These operations extend to matrices which allow them to be part of vector spaces.
• Linear Transformations: Matrices can represent functions from one vector space to another, enabling them to transform vectors while preserving vector addition and scalar multiplication properties.

Understanding the fundamental properties of matrices provides a foundation to explore other linear algebra concepts, such as null spaces.

Null Space

The null space, or kernel, of a matrix \(A\) is the set of all vectors \(\mathbf{x}\) such that \(A\mathbf{x} = \mathbf{0}\). In this exercise, the vector \(\mathbf{y}\) is in the null space of each matrix \(A_i\). This means when the matrix acts on \(\mathbf{y}\), the result is the zero vector.
The null space has significant importance in understanding the solutions to linear systems. It can help reveal the internal structure of a matrix.
Key points about null spaces:

• Nullity: The dimension of the null space, called the nullity, indicates the number of solutions that result in zero. A higher nullity often means more freedom or solutions.
• Linearly Dependent Vectors: If a vector is in the null space, it indicates some level of dependency of rows or columns in the matrix.

In our exercise, because \(\mathbf{y} eq \mathbf{0}\) and still resides in the null space of all matrices \(A_i\), it provides critical insight into their inability to span \(\mathbf{M}_{nn}\).

Linear Independence

Vectors from a set are linearly independent if no vector in the set can be written as a combination of the others. In terms of matrices, this concept is extended to understand if transformations can span space entirely.
For the given exercise, we suspect a contradiction when we assert the matrices \(\{A_1, A_2, \ldots, A_m\}\) span \(\mathbf{M}_{nn}\). Each \(A_i\) shares a common vector \(\mathbf{y}\) in their null space, indicating dependency.
Important characteristics of linear independence:

• Span: Independent vectors are needed to span a high-dimensional space like \(\mathbf{M}_{nn}\). Dependency implies a restriction in span.
• Basis: A minimal complete set of independent vectors forms a basis for a vector space. If \(\{A_1, A_2, \ldots, A_m\}\) were a basis, every possible \(n \times n\) transformation could be rendered impossible due to this common null space.

Because of this common-null-space trait, \(\{A_1, A_2, \ldots, A_m\}\) cannot fully express every matrix, preventing them from spanning \(\mathbf{M}_{nn}\).

Vector Spaces

In linear algebra, a vector space is a collection of vectors that can be added together and multiplied by scalars, with operations satisfying specific axioms. Vector spaces provide the backdrop for matrix operations, allowing us to apply abstract concepts to practical problems.
In the context of this exercise, the set of all \(n \times n\) matrices \(\mathbf{M}_{nn}\) forms a vector space, where matrices are considered vectors within this space.
Core aspects of vector spaces to understand here:

• Closure: Adding two matrices or scaling by a factor remains within the space, showing closure under addition and multiplication.
• Dimension: Represented by a basis, the dimension tells how many independent directions exist in the space. Matrices \(\{A_1, A_2, \ldots, A_m\}\) cannot form a complete set if they rely on a common vector like \(\mathbf{y}\).

The inability of these matrices to express all elements of \(\mathbf{M}_{nn}\) directly affects how they interact with the entirety of vector transformations, emphasizing critical linear algebraic principles.