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Chapter 7: Problem 38

Prove Theorem 7.13: Let \(\left\\{e_{1}, \ldots, e_{n}\right\\}\) be an orthonormal basis of an inner product space \(V .\) Let \(P=\left[a_{i j}\right]\) be an orthogonal matrix. Then the following \(n\) vectors form an orthonormal basis for \(V:\) \\[e_{i}^{\prime}=a_{1 i} e_{1}+a_{2 i} e_{2}+\cdots+a_{n i} e_{n}, \quad i=1,2, \ldots, n\\]

Short Answer

Expert verified
We have proven Theorem 7.13 by showing that the given set of vectors \({e_1',\dots, e_n'}\) are orthogonal, of unit length, and form a basis for the inner product space V. This was done by computing their inner product, verifying their length, and checking they form a basis.

Step by step solution

01

Compute the inner product of each pair of vectors and check for orthogonality

Let's calculate the inner product of arbitrary vectors \(e_i'\) and \(e_j'\): \[ \begin{aligned} (e_i', e_j') &= \left(\sum_{k=1}^n a_{ki} e_k, \sum_{l=1}^n a_{lj} e_l\right) \\ &= \sum_{k=1}^n \sum_{l=1}^n a_{ki} a_{lj} (e_k, e_l) \end{aligned} \] Since \(e_k\) and \(e_l\) form an orthonormal basis, we have \((e_k, e_l) = \delta_{kl}\), where \(\delta_{kl}\) is Kronecker delta function: \[ \delta_{kl}=\left\{ \begin{array}{ll} 1, & k=l \\ 0, & k \neq l \end{array} \right. \] So, the inner product will become: \[ (e_i', e_j') = \sum_{k=1}^n a_{ki} a_{kj} \delta_{kl} = \sum_{k=1}^n a_{ki} a_{kj} \] Now, we'll use the property of orthogonal matrices: \(P^T P = I\), where \(I\) is the identity matrix. In index notation, \(a_{ik} a_{jk} = \delta_{ij}\). Thus the inner product becomes: \[ (e_i', e_j') = \delta_{ij}, \] which implies that \(e_i'\) and \(e_j'\) are orthogonal if \(i \neq j\).
02

Verify that each vector is of unit length

To show that the vectors are of unit length, we will consider the inner product of \(e_i'\) with itself: \[ (e_i', e_i') = \sum_{k=1}^n a_{ki} a_{ki} = \sum_{k=1}^n a_{ki}^2 \] Using the orthogonal matrix property, this will yield: \[ (e_i', e_i') = \delta_{ii} = 1 \] Hence, each vector \(e_i'\) is of unit length.
03

Confirm that the vectors form a basis for V

Since we have shown that the vectors \(e_i'\) are orthogonal and of unit length, and there are exactly \(n\) of them, they form an orthonormal basis of the same size as the given orthonormal basis. Thus, the set of vectors \({e_1',\dots, e_n'}\) forms an orthonormal basis for the inner product space V. In conclusion, we have proven Theorem 7.13 by showing that the given set of vectors \({e_1',\dots, e_n'}\) are orthogonal, of unit length, and form a basis for the inner product space V.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inner Product Space
An inner product space is a vector space equipped with an inner product. This mathematical concept allows for the generalization of the dot product. The inner product provides a way to multiply vectors together, giving an output that is a scalar. This scalar value is frequently used to determine angles, lengths, and orthogonality. In an inner product space, the inner product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\), denoted by \((\mathbf{u}, \mathbf{v})\), satisfies the following axioms:
  • Conjugate Symmetry: \((\mathbf{u}, \mathbf{v}) = \overline{(\mathbf{v}, \mathbf{u})}\)
  • Linearity in the first argument: \((a\mathbf{u} + b\mathbf{w}, \mathbf{v}) = a(\mathbf{u}, \mathbf{v}) + b(\mathbf{w}, \mathbf{v})\) for any scalars \(a\) and \(b\)
  • Positive-Definiteness: \((\mathbf{u}, \mathbf{u}) \geq 0\) with equality if and only if \(\mathbf{u} = 0\)
These properties allow for a detailed understanding of vector behavior, making inner product spaces essential in various fields such as physics and engineering.
Orthogonal Matrix
An orthogonal matrix plays a crucial role in linear algebra, particularly when dealing with transformations. This type of matrix is characterized by the property that its transpose equals its inverse. Hence, for an orthogonal matrix \(P\), you have:\[ P^T P = I \]where \(I\) represents the identity matrix. This identity suggests that the rows and columns of \(P\) are orthonormal sets of vectors. Properties of orthogonal matrices include:
  • Preservation of length: Transforming a vector \(\mathbf{v}\) with an orthogonal matrix \(P\) does not change its length. If \(\mathbf{v}' = P\mathbf{v}\), then \(||\mathbf{v}'|| = ||\mathbf{v}||\).
  • Preservation of angles: Because they preserve the dot product, orthogonal transformations also preserve angles between vectors.
  • Determinant: An orthogonal matrix has a determinant of either +1 or -1, which signifies its transformation properties.
Understanding these properties is essential when exploring the nature of rotations and reflections in geometry.
Kronecker Delta Function
The Kronecker delta function is a mathematical tool that simplifies expressions involving sums of products. This function is extremely useful in linear algebra and theoretical physics. It is defined as:\[ \delta_{kl} = \begin{cases} 1, & \text{if } k = l \0, & \text{if } k eq l \end{cases} \]The function acts like a switch, turning on (value 1) when its indices match and turning off (value 0) when they differ. In the context of inner product spaces, the Kronecker delta is often used to express orthonormality conditions.For instance, for the orthonormal basis \(\{e_k\}\) in a vector space, we have \((e_k, e_l) = \delta_{kl}\). This means that the vectors \(e_k\) and \(e_l\) are orthogonal unless \(k = l\), in which case they are perpendicular and have length 1. This paradigmatic property forms the backbone of orthonormal sets and simplifies computations in orthogonality checks.
Linear Algebra Theorem
Linear algebra is full of pivotal theorems that help in understanding vector spaces and transformations. Theorem 7.13 from the problem highlights a fundamental construction related to orthonormal bases and orthogonal matrices.The theorem essentially states that if \(\{e_1, \ldots, e_n\}\) is an orthonormal basis of an inner product space \(V\), and \(P\) is an orthogonal matrix, then the transformed vectors \(e_{i}' = \sum_{k=1}^{n} a_{ki} e_{k}\) form another orthonormal basis. This concept plays a crucial role when discussing transformation properties and simplifications offered by orthogonal matrices.Key takeaway points from this theorem include:
  • Every orthogonal transformation (given by an orthogonal matrix) of an orthonormal basis remains an orthonormal basis in the same space.
  • The preservation of orthonormality simplifies the computational complexity in numerical methods and algorithms.

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Most popular questions from this chapter

A complex matrix \(A\) is unitary if it is invertible and \(A^{-1}=A^{H}\). Alternatively, \(A\) is unitary if its rows (columns) form an orthonormal set of vectors (relative to the usual inner product of \(\mathbf{C}^{n}\) ). Find a unitary matrix whose first row is: (a) a multiple of \((1,1-i)\); (b) a multiple of \(\left(\frac{1}{2}, \frac{1}{2} i, \frac{1}{2}-\frac{1}{2} i\right)\).

Prove Theorem 7.4: Let \(W\) be a subspace of \(V\). Then \(V=W \oplus W^{\perp}\). By Theorem 7.9, there exists an orthogonal basis \(\left\\{u_{1}, \ldots, u_{r}\right\\}\) of \(W\), and by Theorem \(7.10\) we can extend it to an orthogonal basis \(\left\\{u_{1}, u_{2}, \ldots, u_{n}\right\\}\) of \(V\). Hence, \(u_{r+1}, \ldots, u_{n} \in W^{\perp}\). If \(v \in V\), then \(v=a_{1} u_{1}+\cdots+a_{n} u_{n}\), where \(a_{1} u_{1}+\cdots+a_{r} u_{r} \in W\) and \(a_{r+1} u_{r+1}+\cdots+a_{n} u_{n} \in W^{\perp}\) Accordingly, \(V=W+W^{\perp}\). On the other hand, if \(w \in W \cap W^{\perp}\), then \(\langle w, w\rangle=0\). This yields \(w=0\). Hence, \(W \cap W^{\perp}=\\{0\\}\). The two conditions \(V=W+W^{\perp}\) and \(W \cap W^{\perp}=\\{0\\}\) give the desired result \(V=W \oplus W^{\perp}\). Remark: Note that we have proved the theorem for the case that \(V\) has finite dimension. We remark that the theorem also holds for spaces of arbitrary dimension.

Let \(V\) be a complex inner product space. Verify the relation \\[\left\langle u, a v_{1}+b v_{2}\right\rangle=\bar{a}\left\langle u, v_{1}\right\rangle+\bar{b}\left\langle u, v_{2}\right\rangle\\]

Find an orthogonal matrix \(P\) whose first row is \(u_{1}=\left(\frac{1}{3}, \frac{2}{3}, \frac{2}{3}\right)\) First find a nonzero vector \(w_{2}=(x, y, z)\) that is orthogonal to \(u_{1}-\) that is, for which \\[0=\left\langle u_{1}, w_{2}\right\rangle=\frac{x}{3}+\frac{2 y}{3}+\frac{2 z}{3}=0 \quad \text { or } \quad x+2 y+2 z=0\\]

Find the Fourier coefficient \(c\) and projection \(c w\) of \(v\) along \(w,\) where (a) \(\quad v=(2,3,-5)\) and \(w=(1,-5,2)\) in \(\mathbf{R}^{3}\). (b) \(v=(1,3,1,2)\) and \(w=(1,-2,7,4)\) in \(\mathbf{R}^{4}\). (c) \(v=t^{2}\) and \(w=t+3\) in \(\mathbf{P}(t),\) with inner product \(\langle f, g\rangle=\int_{0}^{1} f(t) g(t) d t\). (d) \(v=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]\) and \(w=\left[\begin{array}{ll}1 & 1 \\ 5 & 5\end{array}\right]\) in \(\mathbf{M}=\mathbf{M}_{2,2},\) with inner product \(\langle A, B\rangle=\operatorname{tr}\left(B^{T} A\right)\).
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