Let 𝐮1, …, 𝐮n and 𝐮̂1, …, 𝐮̂n be orthonormal bases of an...

Let $\mathbf{u}_1, \ldots, \mathbf{u}_n$ and $\hat{\mathbf{u}}_1, \ldots, \hat{\mathbf{u}}_n$ be orthonormal bases of an inner product space $V$. Prove that $\widehat{\mathbf{u}}_i=\sum_{j=1}^n q_{i j} \mathbf{u}_j$ for $i=1, \ldots, n$, where $Q=\left(q_{i j}\right)$ is an orthogonal matrix.

Key Concepts

Orthogonal Matrix

An orthogonal matrix is a square matrix whose rows and columns are orthonormal vectors, meaning that multiplying the matrix by its transpose yields the identity matrix. Orthogonal matrices represent rotations or reflections in Euclidean space and preserve the inner product, hence, distances and angles, which is why they are significant in change-of-basis transformations in inner product spaces.

Inner Product Space

An inner product space is a vector space equipped with an inner product, which is a function that assigns a scalar to each pair of vectors and satisfies properties such as linearity, symmetry, and positive-definiteness. This structure allows for defining concepts of length, angle, and orthogonality, which are foundational to understanding geometric properties in both finite and infinite dimensional spaces.

Orthonormal Basis

An orthonormal basis of an inner product space is a set of vectors that are mutually orthogonal and each of unit length. This property simplifies computations in the space, as every vector can be represented uniquely as a linear combination of basis vectors, and the coefficients are directly determined by the inner products with the basis vectors.

Change of Basis Matrix

A change of basis matrix is used to transform the coordinates of vectors from one basis to another. In the context of inner product spaces, when both bases are orthonormal, the matrix that converts one basis representation to another retains special algebraic properties, making the transformation process more straightforward.

Transcript

00:01 We're asked to prove theorem 713 from this chapter.

00:07 Recall what this theorem says.

00:08 It says that if we're given an orthomormal basis, e1 through e .n, of a vector space v.

00:27 And in fact, not just a vector space, but an inner product space.

00:38 And we're given...

00:44 An orthogonal matrix p with entries a .i .j.

01:02 Then it follows that the n vectors, ei, prime, which is a1i1, plus a2ie2, all the way up to a, n, i, the n, where i ranges from one up to n.

01:33 Well, these n vectors, these form an orthonormal basis.

01:43 No, no, he's doing a comeback out.

01:44 I follow him on instagram.

01:46 Yeah, of course he'd say that on instagram.

01:49 For v also.

01:51 They all fucking home, i guess.

01:53 Oh, he's not into coke or yachts anymore.

01:56 He's like out his priorities.

01:58 But he does like smoke weed in the mall.

02:01 That's like his...

02:01 Prove this statement, well, because the set ei is an orthonormal set, it follows by a previous exercise at the inner products of ei prime and ej prime, which these are linear combinations of the ei.

02:26 Well, hey guys, these are the sum of the products of corresponding coefficients.

02:35 So, a1i times a1j plus a2i times a2j, all the way up to plus a.

02:48 A .n .i times a .n .j...

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