Question

In each case either show that the statement is true or give an example showing that it is false. a. Every independent set in \(\mathbb{R}^{n}\) is orthogonal. b. If \(\\{\mathbf{x}, \mathbf{y}\\}\) is an orthogonal set in \(\mathbb{R}^{n},\) then \(\\{\mathbf{x}, \mathbf{x}+\mathbf{y}\\}\) is also orthogonal. c. If \(\\{\mathbf{x}, \mathbf{y}\\}\) and \(\\{\mathbf{z}, \mathbf{w}\\}\) are both orthogonal in \(\mathbb{R}^{n}\), then \(\\{\mathbf{x}, \mathbf{y}, \mathbf{z}, \mathbf{w}\\}\) is also orthogonal. d. If \(\left\\{\mathbf{x}_{1}, \mathbf{x}_{2}\right\\}\) and \(\left\\{\mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\right\\}\) are both or- thogonal and \(\mathbf{x}_{i} \cdot \mathbf{y}_{j}=0\) for all \(i\) and \(j,\) then \(\left\\{\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{y}_{1}, \mathbf{y}_{2}, \mathbf{y}_{3}\right\\}\) is orthogonal. e. If \(\left\\{\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{n}\right\\}\) is orthogonal in \(\mathbb{R}^{n}\), then \(\mathbb{R}^{n}=\operatorname{span}\left\\{\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{n}\right\\}\) f. If \(\mathbf{x} \neq \mathbf{0}\) in \(\mathbb{R}^{n},\) then \(\\{\mathbf{x}\\}\) is an orthogonal set.

Short answer

Statements (d) and (f) are true; others are false.

Step-by-step solution

Understanding Independence and Orthogonality

An independent set in \(\mathbb{R}^{n}\) means no vector in the set can be written as a linear combination of others. An orthogonal set means that any two vectors in the set are perpendicular, i.e., their dot product is zero.

Example for False Statement (a)

Consider set \(\{(1,0), (0,1), (1,1)\}\) in \(\mathbb{R}^{2}\). This set is independent because no vector can be written as a combination of the others, but it is not orthogonal since \((1,0)\cdot(1,1) eq 0\). Hence, statement (a) is false.

Verifying Statement (b)

For an orthogonal set \(\{\mathbf{x}, \mathbf{y}\}\), \(\mathbf{x} \cdot \mathbf{y} = 0\). Now, check the orthogonality of \(\{\mathbf{x}, \mathbf{x} + \mathbf{y}\}\). Calculate, \(\mathbf{x} \cdot (\mathbf{x} + \mathbf{y}) = \mathbf{x} \cdot \mathbf{x} + \mathbf{x} \cdot \mathbf{y} = \|\mathbf{x}\|^2 + 0\), which is non-zero. Thus, \{\mathbf{x}, \mathbf{x} + \mathbf{y}\}\ is not orthogonal. Statement (b) is false.

Verifying Statement (c)

For \(\{\mathbf{x}, \mathbf{y}\}\) and \(\{\mathbf{z}, \mathbf{w}\}\) being orthogonal, you need \(\mathbf{x} \cdot \mathbf{y} = 0\) and \(\mathbf{z} \cdot \mathbf{w} = 0\). However, without knowing relationships between \(\mathbf{x}, \mathbf{y}\) and \(\mathbf{z}, \mathbf{w}\), you can't assume \(\mathbf{x} \cdot \mathbf{z} = 0\) or \(\mathbf{y} \cdot \mathbf{w} = 0\). Therefore, \{\mathbf{x}, \mathbf{y}, \mathbf{z}, \mathbf{w}\}\ might not be orthogonal. Statement (c) is false.

Verifying Statement (d)

Given \(\{\mathbf{x}_1, \mathbf{x}_2\}\) and \(\{\mathbf{y}_1, \mathbf{y}_2, \mathbf{y}_3\}\) are orthogonal, and \(\mathbf{x}_i \cdot \mathbf{y}_j = 0\), when combined, these conditions ensure all pairs in the new set are orthogonal. Statement (d) is true.

Understanding Span and Orthogonal (e)

Orthogonality doesn't imply spanning \(\mathbb{R}^{n}\). An orthogonal set spanning \(\mathbb{R}^{n}\) needs \(n\) linearly independent vectors. For example, \(\{(1,0,0), (0,1,0)\}\) in \(\mathbb{R}^{3}\) is orthogonal but doesn't span \(\mathbb{R}^{3}\). Statement (e) is false.

Verifying Singleton Orthogonal Set (f)

A single non-zero vector \(\{\mathbf{x}\}\) always forms an orthogonal set with itself because we don't require a dot product with another vector. Statement (f) is true.

Key concepts

Vector Independence

In linear algebra, a set of vectors is said to be **linearly independent** if no vector in the set can be written as a linear combination of the others. This characteristic is fundamental because it signals unique contributions from each vector in spanning a vector space.
For example:

• The vectors \( (1, 0) \) and \( (0, 1) \) in \( \mathbb{R}^{2} \) are linearly independent because neither can be expressed using multiples of the other.
• Consider \( (1, 0, 0) \), \( (0, 1, 0) \), and \( (0, 0, 1) \) in \( \mathbb{R}^{3} \); they form a set of linearly independent vectors, representing the standard basis in \( \mathbb{R}^{3} \).

Understanding vector independence is crucial as it underpins the formation of vector spaces through unique and distinct directional components. This differs from orthogonal sets, which focus on angles and dot products.

Orthogonal Sets

Orthogonal sets consist of vectors that are pairwise perpendicular to each other. This means that the dot product between any two distinct vectors in the set is zero.
For instance:

• If \( \mathbf{u} = (1, 0) \) and \( \mathbf{v} = (0, 1) \) in \( \mathbb{R}^{2} \), then these vectors are orthogonal because their dot product \( \mathbf{u} \cdot \mathbf{v} = 0 \).
• Orthogonal sets extend to higher dimensions, like \( \{(1, 0, 0), (0, 1, 0), (0, 0, 1)\} \) in \( \mathbb{R}^{3} \), where each vector is orthogonal to the others.

Orthogonality is helpful for simplifying calculations and understanding geometry in vector spaces. It particularly aids in the construction of orthogonal bases and is intrinsic to techniques like the Gram-Schmidt process.

Dot Product

The **dot product** (also known as the scalar product) of two vectors provides a way to multiply them, yielding a scalar quantity. The result indicates the magnitude of projection of one vector onto another, and it's also integral in checking orthogonality.
For vectors \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \):

• The dot product is calculated as \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \).
• If \( \mathbf{a} \cdot \mathbf{b} = 0 \), the vectors are orthogonal.

The dot product also plays a role in calculating angles between vectors, using the formula \( \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{||\mathbf{a}|| \, ||\mathbf{b}||} \), where \( ||\mathbf{a}|| \) is the magnitude of \( \mathbf{a} \). Knowing how to compute and interpret the dot product is essential for solving many geometrical problems and understanding vector projections.

Span in Vector Spaces

The **span** of a set of vectors in a vector space is all possible linear combinations of these vectors. It defines the subspace that the vectors generate, encapsulating the essence of a particular direction and magnitude in the space.
Consider the set \( \{\mathbf{x}_1, \mathbf{x}_2\} \) in \( \mathbb{R}^{2} \):

• The span of these vectors includes every vector that can be expressed in the form \( a\mathbf{x}_1 + b\mathbf{x}_2 \), where \( a \) and \( b \) are scalars.
• If \( \mathbf{x}_1 = (1, 0) \) and \( \mathbf{x}_2 = (0, 1) \), their span covers the entire \( \mathbb{R}^{2} \) plane.

Diversifying the number of independent vectors in a span increases the dimensionality of the subspace. As such, the span is a gateway to understanding how vector combinations create complex structures and forms within a space.