(a) How many orthonormal bases does ℝ have? (b) What about ℝ^2 ? (c) Does your answer change if

step 1 We're given the vector space M. This is the set of all two by two matrices. Equipped with the in

(a) How many orthonormal bases does ℝ have? (b) What about...

Key Concepts

Independence from Specific Inner Product Choice

While the precise vectors in an orthonormal basis depend on the chosen inner product, any two inner products on a finite-dimensional real vector space are equivalent up to a change of coordinates. Hence, the overall structure and the number of orthonormal bases (whether finite or infinite) remain unchanged under a different inner product.

Degrees of Freedom in Basis Selection

The number of orthonormal bases in a vector space is related to the degrees of freedom available for choosing distinct unit vectors while preserving orthogonality. In one-dimensional spaces there are only finitely many choices (essentially two, corresponding to a vector and its negative), whereas in higher dimensions the continuous range of possible directions leads to an infinite number of orthonormal bases.

Inner Product Space

An inner product space is a vector space equipped with an additional structure called an inner product. This inner product allows one to define lengths (norms) and angles between vectors, which in turn enables the concept and construction of orthogonality, a crucial element for forming orthonormal bases.

Orthonormal Basis

An orthonormal basis of an inner product space is a set of vectors that are both orthogonal to one another and of unit length. This special type of basis simplifies many computations, such as those involving projections and decompositions, and its properties depend on the defined inner product.

Transcript

00:03 We're given the vector space m.

00:09 This is the set of all two by two matrices.

00:15 Equipped with the inner product of ab is the trace of b transpose times a.

00:30 We're asked to show at the set with matrices 1 -0 -0 -0 -0.

00:35 With matrices 1 -0 -0 -0 -1 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 and 0 -0 -1.

00:52 We're asked to show that this set is an orthorn -normal basis for m.

01:01 Well, first of all, i'll call this set set.

01:10 S and i'll call these matrices e1, e2, well instead of doing that, e1, e12, e12, e12, e12, e2, e2, now we know the inner product of e11 with itself.

01:45 It's going to be more convenient to use the alternative.

01:50 Alternative definition that the inner product of a with b, while using the trace, this is also the sum from i equals 1 to 2, sum from j equals 1 to 2 of a .i .j times di .j.

02:10 Therefore, the inner product of e11 with itself is equal to well, we have 1 times 1 plus 0 times 0 plus 0 plus 0 times 0, which is just 1.

02:35 We have the inner product of e11 with e12 is 1 times 0 plus 0 times 0 plus 0 times 0, which is 0 ,000, which is 0.

02:52 The inner product of e11 with e1 or e21 is 1 times 0 plus 0 plus 0 times 1 plus 0 times 1 plus 0 times 0, which is 0.

03:10 And the interproduct of e11 with e22 is 1 times 0 plus 0 plus 0 plus 1 times 0 plus 1 times 1, which is...

03:22 Sorry, 0 times 1, which is 0...

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