Problem 36 - Advanced Engineering Mathematics

Question

Consider a finite orthogonal set of nonzero vectors \(\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right\\}\) in \(R^{n} .\) Discuss: Is this set linearly independent or linearly dependent?

Step-by-Step Solution

Answer

The set is linearly independent.

Step 1: Define Orthogonality

A set of vectors is orthogonal if every pair of distinct vectors in the set is orthogonal, meaning their dot product is zero. For the given set \( \{ \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k \} \), we have \( \mathbf{v}_i \cdot \mathbf{v}_j = 0 \) for all \( i eq j \).

Step 2: Linear Independence Definition

A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. Alternatively, \( \sum_{i=1}^{k} c_i \mathbf{v}_i = \mathbf{0} \) implies that all coefficients \( c_i \) are zero.

Step 3: Use Orthogonality for Independence

In an orthogonal set, each vector \( \mathbf{v}_i \) cannot be expressed as a linear combination of the other vectors due to the zero dot product condition. Hence, a nonzero orthogonal set must be linearly independent.

Step 4: Confirmation Conclusion

Since the given set of vectors is orthogonal and all vectors are nonzero, the set must be linearly independent according to the properties of orthogonal sets.

Key Concepts

Orthogonal Vectors

When learning about vectors, orthogonality is a fascinating concept that crops up quite often, particularly because of its connection to linear independence. Put simply, orthogonal vectors are vectors that meet at right angles to each other. In mathematical terms, this means that their dot product is zero. For example, for any pair of vectors \( \mathbf{v}_i \) and \( \mathbf{v}_j \), the dot product being zero \( \mathbf{v}_i \cdot \mathbf{v}_j = 0 \) tells us they are orthogonal in \( R^n \) space.

This property has significant implications. When dealing with orthogonal vectors, it is important to remember the following:

• Orthogonal vectors do not influence one another in terms of direction within the vector space.
• This lack of influence is what contributes to the vectors' linear independence.

In essence, if you have a set of nonzero orthogonal vectors, you can be certain that this set is linearly independent.

Finite Vector Sets

The understanding of finite vector sets is crucial in linear algebra. A finite vector set means you have a defined number of vectors, as opposed to an infinite set. Typically, these finite sets are easier to handle.

When considering the concept of finite sets, especially one with orthogonal vectors like \( \{ \mathbf{v}_1, \mathbf{v}_2, \, \ldots, \, \mathbf{v}_k \} \), remember:

• The standard operations and properties, such as the geometry of the vectors, are locked by the fixed number of vectors.
• Linear combinations, span, basis calculations, etc., are performed similarly as they would be on an individual vector; however, in a finite context, such operations are limited to the constituents of the set.

The beauty of finite vector sets lies in their manageability and the clear boundaries within which you work, making them an excellent learning model for students delving into vector algebra.

R^n Vector Spaces

The term \( R^n \) refers to an \( n \)-dimensional Euclidean space. When working with vectors, understanding \( R^n \) is essential as it depicts the multi-dimensional space where vectors live. Each vector within \( R^n \) can be represented in a geometrical space with \( n \) components.

In the context of \( R^n \), vectors are operated upon with familiar algebraic rules. Here are key aspects to note:

• The dimension \( n \) refers to the number of coordinates a vector in this space has. For example, in \( R^3 \), vectors have three coordinates \((x, y, z)\).
• Understanding the dimension helps you visualize vector interactions, such as orthogonality or linear independence.
• Since \( R^n \) contains infinite possible vectors, any finite set within it is automatically limited by its size, which simplifies calculations.

In summary, \( R^n \) is your go-to framework for visualizing and solving vector problems, training your ability to manage complex spatial problems in both theoretical and practical scenarios.