Question

If \(V=\operatorname{span}\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n}\right\\}\) and \(\left\langle\mathbf{v}, \mathbf{v}_{i}\right\rangle=\left\langle\mathbf{w}, \mathbf{v}_{i}\right\rangle\) holds for each \(i\). Show that \(\mathbf{v}=\mathbf{w}\)

Short answer

Since \( \langle \mathbf{v}, \mathbf{v}_i \rangle = \langle \mathbf{w}, \mathbf{v}_i \rangle \) for all \( i \), and \( \{ \mathbf{v}_1, \ldots, \mathbf{v}_n \} \) span \( V \), it follows that \( \mathbf{v} = \mathbf{w} \) by linear independence.

Step-by-step solution

Understand the Given Information

We are given that a vector space \( V \) is spanned by a set of vectors \( \{ \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \} \). This means any vector in \( V \) can be written as a linear combination of these vectors. We also have inner product conditions: \( \langle \mathbf{v}, \mathbf{v}_i \rangle = \langle \mathbf{w}, \mathbf{v}_i \rangle \) for each \( i \).

Use the Properties of Inner Product

Recall that for any vectors \( \mathbf{a} \) and \( \mathbf{b} \) in an inner product space, \( \langle \mathbf{a}, \mathbf{b} \rangle = \langle \mathbf{b}, \mathbf{a} \rangle \), and inner products are linear. The condition \( \langle \mathbf{v}, \mathbf{v}_i \rangle = \langle \mathbf{w}, \mathbf{v}_i \rangle \) suggests that the inner products of \( \mathbf{v} \) and \( \mathbf{w} \) with each basis vector \( \mathbf{v}_i \) are equal.

Express Vectors as Linear Combinations

Since \( \mathbf{v} \) and \( \mathbf{w} \) are in the span of \( \{ \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \} \), express \( \mathbf{v} = c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \cdots + c_n \mathbf{v}_n \) and \( \mathbf{w} = d_1 \mathbf{v}_1 + d_2 \mathbf{v}_2 + \cdots + d_n \mathbf{v}_n \).

Equate Inner Products for Basis Vectors

Substitute the linear combinations into the inner product conditions. For each \( i \), we have \( \langle \mathbf{v}, \mathbf{v}_i \rangle = c_1 \langle \mathbf{v}_1, \mathbf{v}_i \rangle + c_2 \langle \mathbf{v}_2, \mathbf{v}_i \rangle + \cdots + c_n \langle \mathbf{v}_n, \mathbf{v}_i \rangle \). Set this equal to the corresponding expression for \( \mathbf{w} \): \( d_1 \langle \mathbf{v}_1, \mathbf{v}_i \rangle + d_2 \langle \mathbf{v}_2, \mathbf{v}_i \rangle + \cdots + d_n \langle \mathbf{v}_n, \mathbf{v}_i \rangle \).

Solve the System of Equations

The equations from Step 4 can be written as a system of linear equations in terms of the coefficients \( c_i \) and \( d_i \). Since all inner products are equal, each equation implies \( c_i = d_i \). Verify each coefficient pair equates which ensures \( \mathbf{v} = \mathbf{w} \).

Conclude with Linear Independence

Because the vectors \( \{ \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \} \) span \( V \) and are linearly independent, having equal inner product results leads directly to the conclusion that \( c_i = d_i \) for all \( i \), meaning \( \mathbf{v} = \mathbf{w} \).

Key concepts

Vector Spaces

In the world of linear algebra, a vector space is a fundamental structure that allows us to perform operations like addition and scalar multiplication within a defined set of vectors. A vector space, such as the given space \( V = \operatorname{span}\{\mathbf{v}_{1}, \mathbf{v}_{2}, \, \ldots, \, \mathbf{v}_{n}\}\), is spanned by its basis vectors. This means every single vector in \( V \) can be represented as a linear combination of these basis vectors.

• Basis vectors are crucial in forming the vector space as they define its dimensions.
• For example, if a vector \( \mathbf{u} \) lies in \( V \), it can be expressed with coefficients of the basis vectors.
• This representation is unique if the basis itself is linearly independent.

The importance of vector spaces lies in their applicability across different fields, which include physics, engineering, and computer science, where they help model and solve problems.

Inner Product

The concept of an inner product is an extension to vector spaces that introduces the notion of angle and length. Essentially, it is a mathematical tool that helps define and measure the geometric attributes of vectors. In our exercise, the inner product condition \( \langle \mathbf{v}, \mathbf{v}_i \rangle = \langle \mathbf{w}, \mathbf{v}_i \rangle \) reflects an important role it plays in determining the relationships between vectors.

• Inner products are often denoted by \( \langle \mathbf{a}, \mathbf{b} \rangle \), which may represent the dot product in euclidean spaces.
• This operation is linear, allowing it to distribute across vector addition and scalar multiplication.
• It is also symmetric, which means \( \langle \mathbf{a}, \mathbf{b} \rangle = \langle \mathbf{b}, \mathbf{a} \rangle \).

By comparing inner products, as in the given problem, we can deduce equivalencies between vectors because inner products encapsulate their "projections" against a common reference, here the basis vector \( \mathbf{v}_i \).

Linear Combinations

A linear combination involves taking several vectors and combining them using specific scalar coefficients to form another vector. In the exercise, we're dealing with expressing vectors \( \mathbf{v} \) and \( \mathbf{w} \) as linear combinations of a set of basis vectors like \( \mathbf{v} = c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \cdots + c_n \mathbf{v}_n \).Key aspects to remember about linear combinations include:

• The coefficients \( c_1, c_2, \ldots, c_n \) quantify how much of each basis vector is needed to construct another vector.
• These combinations allow us to describe any vector within vector space \( V \) in terms of its basis.
• The concept is powerful for understanding transformations and movements within the vector space.

Using these linear combinations and setting their conditions equal is crucial in our problem as it leads to comparing the underlying components.

Linear Independence

The notion of linear independence is pivotal in determining the uniqueness and behavior of basis vectors within a vector space. A set of vectors \( \{ \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \} \) is linearly independent if no vector can be written as a linear combination of the others. This ensures that there are no redundant vectors in the span.

• If a vector space is spanned by linearly independent vectors, each vector's representation is unique.
• Linear independence implies each \( c_i = d_i \). This equality is vital in concluding our exercise, allowing us to prove \( \mathbf{v} = \mathbf{w} \).
• This property maintains the integrity of vector expressions and remains fundamental in solving linear systems.

Knowing that the basis vectors in our exercise are linearly independent assures us that the coefficients are indeed equal, which leads to equivalent vectors \( \mathbf{v} \) and \( \mathbf{w} \).