Question

If \(\mathbb{R}^{n}=\operatorname{span}\left\\{\mathbf{x}_{1}, \ldots, \mathbf{x}_{m}\right\\}\) and \(\mathbf{x} \cdot \mathbf{x}_{i}=0\) for all \(i,\) show that \(\mathbf{x}=0 .[\) Hint : Show \(\|\mathbf{x}\|=0 .]\)

Short answer

\(\mathbf{x} = 0\) because it is orthogonal to a spanning set of \(\mathbb{R}^n\).

Step-by-step solution

Understand the Given Information

We are given that the n-dimensional real space \(\mathbb{R}^n\) is spanned by the vectors \(\{\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_m\}\). Additionally, it is given that \(\mathbf{x} \cdot \mathbf{x}_i = 0\) for all \(i\), meaning \(\mathbf{x}\) is orthogonal to each of the spanning vectors.

Express the Vector \(\mathbf{x}\) Using the Span

Since \(\mathbb{R}^n\) is spanned by \(\{\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_m\}\), any vector \(\mathbf{y} \in \mathbb{R}^n\), including \(\mathbf{x}\), can be expressed as a linear combination of these vectors: \(\mathbf{x} = c_1\mathbf{x}_1 + c_2\mathbf{x}_2 + \ldots + c_m\mathbf{x}_m\), where \(c_1, c_2, \ldots, c_m\) are real coefficients.

Use Orthogonality Property

Given that \(\mathbf{x} \cdot \mathbf{x}_i = 0\) for all \(i\), substitute the expression from Step 2: \((c_1\mathbf{x}_1 + c_2\mathbf{x}_2 + \ldots + c_m\mathbf{x}_m) \cdot \mathbf{x}_i = 0\). This simplifies to \(c_1(\mathbf{x}_1 \cdot \mathbf{x}_i) + c_2(\mathbf{x}_2 \cdot \mathbf{x}_i) + \ldots + c_m(\mathbf{x}_m \cdot \mathbf{x}_i) = 0\) for each \(i\).

Solve Linear System of Equations

The inner product \(\mathbf{x}_j \cdot \mathbf{x}_i\) between any two spanning vectors forms a system of linear equations based on the coefficients \(c_1, c_2, \ldots, c_m\). Since you have \(\mathbf{x} \cdot \mathbf{x}_i = 0\) for all \(i\), and the span of \(\{\mathbf{x}_1, \ldots, \mathbf{x}_m\}\) covers the entire space, it follows that this system of equations can only be satisfied if all coefficients \(c_i = 0\).

Conclude that \(\mathbf{x} = 0\)

Since all coefficients \(c_1, c_2, \ldots, c_m\) are zero, the expression for \(\mathbf{x}\) simplifies to \(\mathbf{x} = 0\mathbf{x}_1 + 0\mathbf{x}_2 + \ldots + 0\mathbf{x}_m = \mathbf{0}\). Thus, \(\mathbf{x} = \mathbf{0}\) as all components are zero.

Key concepts

Orthogonality

Orthogonality in the realm of linear algebra refers to the concept where two vectors are perpendicular to each other. This can be mathematically expressed by the dot product of the two vectors. If two vectors \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal, then their dot product is zero, i.e., \( \mathbf{a} \cdot \mathbf{b} = 0 \).

Orthogonality is a crucial component in understanding vector spaces as it provides a way to identify when vectors contribute independent directions. In simpler terms, orthogonal vectors pull in different directions entirely, like how the x and y axes in a coordinate plane are perpendicular.

• Orthogonality ensures vectors are not redundant.
• Orthogonal vectors are essential in concepts like projections and in defining bases with desirable properties.
• When solving problems in linear algebra, verifying orthogonality helps in revealing linear independence among vectors.

Given a set of vectors that span a space, a vector orthogonal to each of the spanning vectors provides no new direction and hence contributes a zero or null effect, which is key to solving exercises involving zero solutions.

Vector Spaces

A vector space is a collection of vectors that can be added together and multiplied by scalars – the elements of some field, typically the real numbers, \( \mathbb{R} \). It is defined by specific properties such as closure under addition and scalar multiplication, having a zero vector, and containing additive inverses.

In linear algebra, vector spaces provide a framework for studying linear equations and transformations. They give a structured way to talk about and analyze vectors we encounter in different dimensions, be it 2D vectors on a plane or \( n \)-dimensional vectors.

• Vectors in the space can be manipulated mathematically yet still remain within the space.
• Every vector in a vector space can be described as a linear combination of a basis – a foundational subset of the space.
• Vector spaces are fundamental in defining solutions to linear systems, transformations, and more.

In the context of the exercise, the n-dimensional real space \( \mathbb{R}^n \) is spanned by a set of vectors. This means any vector in this space can be represented as a combination of these spanning vectors, adhering to the rules of the vector space. It acts like a 'playground' where vectors and operations can be explored freely yet systematically.

Linear Combinations

Linear combinations are an integral part of working with vectors. They involve creating new vectors by summing scalar multiples of existing vectors. Consider vectors \( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_m \), a linear combination of these is formed as \( a_1\mathbf{v}_1 + a_2\mathbf{v}_2 + \ldots + a_m\mathbf{v}_m \), where \( a_1, a_2, \ldots, a_m \) are scalars.

Linear combinations are useful for:

• Constructing solutions to systems of linear equations.
• Understanding how vectors relate to each other within a space.
• Determining whether a set of vectors spans a space or is linearly independent.

In the exercise, recognizing \( \mathbb{R}^n \) as spanned by an orthogonally complete set of vectors allows us to express any vector, including \( \mathbf{x} \), through linear combinations of these spanning vectors. If \( \mathbf{x} \) is orthogonal to all spanning directions, it necessarily implies zero coefficients, leading directly to \( \mathbf{x} = \mathbf{0} \), as linear combinations in this scenario do not introduce a new direction.