Question

Given a Euclidean space \(R\), let \(\varphi_{1}, \varphi_{2}, \ldots, \varphi_{k}, \ldots\) be an orthonormal basis in \(R\) and \(\mathbf{f}\) an arbitrary element of \(R\). Prove that the element $$ f-\sum_{k=1}^{n} a_{k} \varphi_{k} $$ is orthogonal to all linear combinations of the form if and only if $$ \sum_{\mathrm{k}=1} b_{k} \varphi_{k} $$ $$ a_{k}=\left(f, \varphi_{k}\right) \quad(k=1,2, \ldots, n) $$

Short answer

The element is orthogonal if and only if \( a_{k} = (f, \varphi_{k}) \).

Step-by-step solution

Understanding the Problem

We need to prove that the element \( f - \sum_{k=1}^{n} a_{k} \varphi_{k} \) is orthogonal to all linear combinations of \( \sum_{k=1}^{n} b_{k} \varphi_{k} \) if and only if \( a_{k} = (f, \varphi_{k}) \). We will use properties of orthonormal bases to approach this problem.

Expressing the Arbitrary Function

The function \( \mathbf{f} \) can be expressed in terms of an orthonormal basis. In this basis, \( \mathbf{f} = \sum_{k=1}^{n} c_{k} \varphi_{k} + \mathbf{g} \), where \( \mathbf{g} \) is orthogonal to all \( \varphi_{k} \) for \( k = 1, 2, \ldots, n \).

Expressing the Inner Product

The inner product of each basis function with the function \( \mathbf{f} \) is given by \( (f, \varphi_{k}) = c_{k} \) since the orthonormal basis satisfies \( (\varphi_{i}, \varphi_{j}) = \delta_{ij} \).

Defining the Projection

The projection of \( \mathbf{f} \) onto the subspace spanned by \( \{\varphi_{1}, \varphi_{2}, \ldots, \varphi_{n}\} \) is \( \sum_{k=1}^{n} (f, \varphi_{k}) \varphi_{k} \).

Establishing Orthogonality

The orthogonality condition requires that \( \left( f - \sum_{k=1}^{n} a_{k} \varphi_{k}, \sum_{m=1}^{n} b_{m} \varphi_{m} \right) = 0 \). Expanding this, we have \( \left( \sum_{k=1}^{n} [c_{k} - a_{k}] \varphi_{k}, \sum_{m=1}^{n} b_{m} \varphi_{m} \right) = 0 \). Since \( \varphi_{k} \) are orthonormal, this product equals zero if and only if each term \( c_{k} - a_{k} = 0 \).

Conclusion of Proof

From step 4, we know that the orthogonality holds if and only if \( c_{k} = a_{k} \), ensuring that each coefficient in the linear combination is equal to the corresponding projection coefficient \( (f, \varphi_{k}) \). Therefore, \( a_{k} = (f, \varphi_{k}) \).

Key concepts

Euclidean Space

Euclidean space is a mathematical concept most often visualized as multi-dimensional generalizations of the 2D plane and 3D space. When we talk about Euclidean space in the context of linear algebra, we're referring to spaces like -dimensional vectors, where points (or vectors) can be added together or multiplied by scalars. This space is equipped with geometric ideas such as distance and angles, allowing us to apply the familiar rules of geometry.

Euclidean space allows us to define important concepts like distance using the Pythagorean theorem. In this way, any vector in an n-dimensional Euclidean space is a mathematical construct that represents a point. Ultimately, Euclidean space provides the structure and framework on which many algebraic operations are based.

Linear Combinations

In the realm of vector spaces, a linear combination involves creating new vectors by multiplying existing vectors by scalars and then adding the results together. Think of it like a mixing board where you blend volumes of sound; here, you are blending vectors with different weights.

To be more specific, a linear combination of a set of vectors (like our basis vectors ) is given by: \[ \sum_{k=1}^{n} b_{k} \varphi_{k} \] where \( b_k \) are scalars (or coefficients). By changing these scalars, you can "steer" the linear combination in various directions within the vector space. Linear combinations are foundational because they help us build new vectors and work toward solving vector equations. The entire span of a vector space is generated by linear combinations of its basis vectors.

Orthogonality

Orthogonality is a crucial concept when dealing with vectors. Two vectors are orthogonal if their inner product is zero. What this means is that they meet at right angles. This is an extension of the idea of perpendicular lines in the plane.

The significance of orthogonality in vector spaces, especially in the context of orthonormal bases, lies in its simplification of mathematical expressions. When vectors in a set are orthogonal to each other, calculations involving these vectors become much simpler. For example, the inner product or projection of one vector onto another will result in zero if they are orthogonal. This principle can be used in numerous applications such as simplifying systems of equations and performing data decompositions in statistics and engineering.

Inner Product

The inner product, also known as a dot product in familiar 2D or 3D spaces, is a mathematical operation that, given two vectors, returns a scalar. It reflects the "dot" of one vector onto another, capturing a measure of their direction similarity. Mathematically, an inner product can be expressed as: \[ (\mathbf{u}, \mathbf{v}) = \sum_{i=1}^{n} u_i v_i \] for vectors \( \mathbf{u} \) and \( \mathbf{v} \).

In the context of orthonormal bases, the inner product can reveal the component of one vector along the direction of another, showing us how much of the first vector goes in the direction of the second.

• If two vectors are orthogonal, their inner product is zero — they are perfectly perpendicular.
• If they are normalized and orthogonal, they form part of what is known as an orthonormal basis, significantly simplifying mathematical operations like vector projections.

Understanding the inner product is fundamental to unraveling the nature of spaces and spans, forming the groundwork for various fields such as quantum mechanics and statistics.