Question

Let \(A\) be an \(m \times n\) matrix. For which columns b in \(\mathbb{R}^{m}\) is \(U=\left\\{\mathbf{x} \mid \mathbf{x} \in \mathbb{R}^{n}, A \mathbf{x}=\mathbf{b}\right\\}\) a subspace of \(\mathbb{R}^{n} ?\) Support your answer.

Short answer

\(U\) is a subspace only when \(\mathbf{b} = \mathbf{0}\).

Step-by-step solution

Definition of a Subspace

A subspace must satisfy three conditions: (1) it contains the zero vector, (2) it is closed under vector addition, and (3) it is closed under scalar multiplication.

Apply the Zero Vector Condition

For \(U\) to be a subspace of \(\mathbb{R}^{n}\), the zero vector \(\mathbf{0} \in \mathbb{R}^{n}\) must be in \(U\). This means \(A \mathbf{0} = \mathbf{b}\) must hold. Since \(A \mathbf{0} = \mathbf{0}\), we require \(\mathbf{b} = \mathbf{0}\).

Verify Other Subspace Conditions

Once \(\mathbf{b} = \mathbf{0}\), the set \(\{\mathbf{x} \mid A\mathbf{x} = \mathbf{0}\}\) automatically satisfies closure under addition and scalar multiplication conditions, because for any \(\mathbf{u}, \mathbf{v} \in U, A(\mathbf{u} + \mathbf{v}) = A\mathbf{u} + A\mathbf{v} = \mathbf{0} + \mathbf{0} = \mathbf{0}\) and for any scalar \(c, A(c\mathbf{u}) = cA\mathbf{u} = c\mathbf{0} = \mathbf{0}\).

Conclusion Based on Conditions

The analysis shows that \(U\) is a subspace of \(\mathbb{R}^{n}\) if and only if \(\mathbf{b} = \mathbf{0}\). Only in this case, all required conditions are satisfied.

Key concepts

Matrix Equation

A matrix equation is a mathematical representation where a matrix, typically denoted as \(A\), is multiplied by a vector, \(\mathbf{x}\), to produce another vector, \(\mathbf{b}\), i.e., \(A\mathbf{x} = \mathbf{b}\). This setup is pivotal for understanding solutions in the context of linear algebra, where we seek vectors \(\mathbf{x}\) that satisfy this relationship. In our exercise, the matrix equation is integral to defining the set \(U\). This set encompasses all possible solutions \(\mathbf{x}\) such that when multiplied by \(A\), the result is the vector \(\mathbf{b}\). Thus, solving a matrix equation essentially involves finding these vectors \(\mathbf{x}\) that satisfy the equation, ultimately defining the characteristics of the solution set \(U\). Understanding the properties of matrix equations helps determine when such a set is a subspace, as it connects the nature of \(\mathbf{b}\) with vector relations.

Zero Vector

The zero vector, often denoted as \(\mathbf{0}\), is a unique vector wherein all its entries are zero. In the context of vector spaces, the presence of a zero vector is a fundamental characteristic that helps decide the structure and nature of a subspace. For the set \(U\) to qualify as a subspace, it must include the zero vector. This is because a subspace is, by definition, required to always contain the zero vector. Referring back to our matrix equation \(A\mathbf{x} = \mathbf{b}\), for \(U\) to be a subspace, it must inherently include the vector solution where \(\mathbf{x} = \mathbf{0}\). This condition is only met when \(A\mathbf{0} = \mathbf{b}\), which simplifies to \(\mathbf{b} = \mathbf{0}\). Thus, the zero vector condition guides us in determining the form of \(\mathbf{b}\) such that \(U\) is indeed a valid subspace.

Closure Under Addition

Closure under addition is one of the core conditions that a set \(U\) must satisfy to be considered a subspace. This property dictates that adding any two vectors from \(U\) results in another vector that is still within \(U\). For instance, if \(\mathbf{u}\) and \(\mathbf{v}\) are vectors in \(U\), then the sum \(\mathbf{u} + \mathbf{v}\) should also be in \(U\). In examining our matrix equation setup \(A\mathbf{x} = \mathbf{b}\), the closure under addition is inherently fulfilled when \(\mathbf{b} = \mathbf{0}\). At this point, any two solutions \(\mathbf{u}\) and \(\mathbf{v}\) both satisfy \(A\mathbf{u} = \mathbf{0}\) and \(A\mathbf{v} = \mathbf{0}\), thus their sum \(A(\mathbf{u} + \mathbf{v}) = A\mathbf{u} + A\mathbf{v} = \mathbf{0} + \mathbf{0} = \mathbf{0}\) remains valid. Therefore, closure under addition is automatically provided, affirming \(U\)'s ability to satisfy this subspace requirement.

Scalar Multiplication

Scalar multiplication is another critical property of vector spaces and subspaces. It stipulates that multiplying any vector in the set \(U\) by a scalar should yield another vector that is still part of \(U\). This means for any vector \(\mathbf{u} \in U\) and any scalar \(c\), the vector \(c\mathbf{u}\) should also belong to \(U\). In the case of the matrix equation \(A\mathbf{x} = \mathbf{b}\), for \(\mathbf{b} = \mathbf{0}\), any vector \(\mathbf{u}\) that satisfies \(A\mathbf{u} = \mathbf{0}\) will still satisfy the equation when multiplied by any scalar \(c\), as \(A(c\mathbf{u}) = cA\mathbf{u} = c\mathbf{0} = \mathbf{0}\). This demonstrates that scalar multiplication holds, as each resulting vector remains within \(U\) when \(\mathbf{b} = \mathbf{0}\). Scalar multiplication together with closure under addition and the zero vector forms the bedrock of the subspace verification.