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Magazine Chapter 3: Problem 5
If \(\left(e_{k}\right)\) is an orthonormal sequence in an inner product space \(X\), and \(x \in X\), show that \(x-y\) with \(y\) given by $$ y=\sum_{k=1}^{n} \alpha_{k} e_{k} \quad \alpha_{k}=\left\langle x, e_{k}\right\rangle $$ is orthogonal to the subspace \(Y_{n}=\operatorname{span}\left\\{e_{t}, \cdots e_{n}\right\\}\)
Short Answer
Expert verified
The vector \(x-y\) is orthogonal to the subspace \(Y_n\).
Step by step solution
01
Understanding the Orthonormal Sequence
Since \((e_k)\) is an orthonormal sequence in the inner product space \(X\), it means for any two elements \(e_i\) and \(e_j\), \(\langle e_i, e_j \rangle = 0\) if \(i eq j\) and \(\langle e_i, e_i \rangle = 1\). This forms a basis for the subspace \(Y_n\).
02
Define the Vector y and the Subspace
We are given \( y = \sum_{k=1}^{n} \alpha_{k} e_{k} \) where \( \alpha_k = \langle x, e_k \rangle \). The subspace \( Y_n = \operatorname{span} \{ e_1, e_2, \ldots, e_n \} \) is spanned by the orthonormal sequence \(e_k\).
03
Write x-y
Calculate \( x - y \). By substituting the expression for \(y\), you get \( x - y = x - \sum_{k=1}^{n} \langle x, e_k \rangle e_k \).
04
Show Orthogonality to Y_n
To prove \(x-y\) is orthogonal to \(Y_n\), consider any element \(z = \sum_{k=1}^{n} \beta_k e_k \) from \( Y_n \). We need \( \langle x-y, z \rangle = 0\).
05
Calculate Inner Product with any z in Y_n
\[\langle x-y, z \rangle = \left\langle x - \sum_{k=1}^{n} \langle x, e_k \rangle e_k, \sum_{j=1}^{n} \beta_j e_j \right\rangle = \langle x, \sum_{j=1}^{n} \beta_j e_j \rangle - \left\langle \sum_{k=1}^{n} \langle x, e_k \rangle e_k , \sum_{j=1}^{n} \beta_j e_j \right\rangle\]
06
Simplify Inner Product Expression
The first term simplifies to \( \sum_{j=1}^{n} \beta_j \langle x, e_j \rangle \). The second term simplifies as follows: \[\left\langle \sum_{k=1}^{n} \langle x, e_k \rangle e_k , \sum_{j=1}^{n} \beta_j e_j \right\rangle = \sum_{k=1}^{n} \sum_{j=1}^{n} \langle x, e_k \rangle \beta_j \langle e_k, e_j \rangle\]Given orthonormality, this reduces to \( \sum_{k=1}^{n} \beta_k \langle x, e_k \rangle\).
07
Conclude the Inner Product Calculation
Plug the expressions back: \[\sum_{j=1}^{n} \beta_j \langle x, e_j \rangle - \sum_{k=1}^{n} \beta_k \langle x, e_k \rangle = 0\]This confirms that \(\langle x-y, z \rangle = 0\) for any \(z\) in \(Y_n\), proving orthogonality.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Orthonormal Sequence
An orthonormal sequence in an inner product space is a sequence of vectors where each pair of different vectors is orthogonal to each other and each vector has a length of one. This is a foundational concept in linear algebra and functional analysis. For instance, in the given problem, the sequence \((e_k)\) is orthonormal in the space \(X\). Here are the critical properties of an orthonormal sequence to remember:
- Each vector \(e_i\) in the sequence satisfies \(\langle e_i, e_i \rangle = 1\), meaning it has a unit length.
- For any two different indices \(i\) and \(j\), the vectors are orthogonal: \(\langle e_i, e_j \rangle = 0\).
- An orthonormal sequence forms an excellent basis for spanning subspaces due to these properties.
Inner Product Space
An inner product space is a vector space coupled with an operation called the inner product. This operation, often denoted by \(\langle.,.\rangle\), takes two vectors and returns a scalar. It generalizes the dot product from Euclidean spaces to more abstract vector spaces. For the problem at hand, the inner product space allows us to use the structure of \(X\) to perform operations like projections and calculate orthogonality.
Key characteristics of inner product spaces include:
Key characteristics of inner product spaces include:
- Linearity: \(\langle ax + by, z \rangle = a\langle x, z \rangle + b\langle y, z \rangle\) for any vectors \(x, y, z\) and scalars \(a, b\).
- Symmetry: \(\langle x, y \rangle = \overline{\langle y, x \rangle}\) (the complex conjugate, if applicable).
- Positive Definiteness: \(\langle x, x \rangle \geq 0\) with equality if and only if \(x = 0\).
Orthogonal Projection
Orthogonal projection refers to projecting a vector onto a subspace such that the resulting vector lies in that subspace and the difference between the original vector and the projection is orthogonal to the subspace. This concept is used in the problem to define the vector \(y\) as an orthogonal projection of \(x\) onto the subspace spanned by the orthonormal sequence \(e_k\).
Here is how orthogonal projection works with orthonormal sequences:
Here is how orthogonal projection works with orthonormal sequences:
- The component of the vector \(x\) along a vector \(e_k\) in the orthonormal basis is given by \(\alpha_k = \langle x, e_k \rangle\).
- The projection \(y\) can be expressed as \(y = \sum_{k=1}^{n} \alpha_k e_k\), where \(\alpha_k\) represents the amount of \(x\) along \(e_k\).
- This construction ensures that \(x - y\) is orthogonal to each \(e_k\), making \(x - y\) orthogonal to the subspace \(Y_n\).
Vector Subspaces
A vector subspace is a subset of a vector space that is also a vector space under the same operations of addition and scalar multiplication. For example, in this problem, \(Y_n\) is a subspace formed by the span of the orthonormal sequence \(\{e_1, e_2, \ldots, e_n\}\).
Subspaces have special properties that are critical in contexts like projection and orthogonality:
Subspaces have special properties that are critical in contexts like projection and orthogonality:
- A subspace must include the zero vector, meaning it contains at least one vector (the simplest possible subspace).
- Any linear combination of vectors within the subspace remains within the subspace.
- The span of any set of vectors creates the smallest subspace containing those vectors.
