Let ℝ^2 have the inner product defined by the positive definite matrix K=( 2 -1 -1

Let ℝ^2 have the inner product defined by the positive...

Key Concepts

Orthonormal Basis

An orthonormal basis is a special type of orthogonal basis in which every basis vector is normalized to have unit length. In other words, the inner product of each basis vector with itself equals one. Orthonormal bases simplify many computations in linear algebra, including the representation of vectors, as the coefficients in a vector’s expansion are simply the inner products with the basis vectors, without needing further normalization adjustments.

Norm in Inner Product Spaces

The norm of a vector in an inner product space is defined as the square root of the inner product of the vector with itself. This norm measures the length or magnitude of the vector. In contexts where the inner product is defined by a matrix, the norm reflects that weighting, and its calculation verifies fundamental relationships like the Pythagorean theorem for orthogonal decompositions.

Vector Decomposition Using Orthogonal Projection

When working in an inner product space with an orthogonal basis, any vector can be uniquely expressed as a linear combination of the basis vectors. The coefficients in this linear combination can be computed using the orthogonality property by projecting the vector onto each basis vector. The formula involves taking the inner product of the vector with the basis vector and dividing by the inner product of the basis vector with itself, which simplifies the process of vector decomposition.

Positive Definite Matrix

A positive definite matrix is a symmetric matrix that yields a positive value when used in a quadratic form with any nonzero vector. This property ensures that the associated inner product leads to a genuine norm where the length of any nonzero vector is positive. It plays a critical role when the inner product is defined via matrix multiplication, providing a framework for measuring distances and angles in the vector space.

Inner Product

An inner product is a function that, given two vectors, returns a scalar while satisfying properties such as linearity, symmetry, and positive-definiteness. In many settings (including finite-dimensional real vector spaces), inner products are used to define lengths (norms) and angles between vectors. In this context, the inner product is defined by a symmetric, positive definite matrix, which allows for generalizing the notion of ‘dot product’ to account for different geometries.

Orthogonal Basis

An orthogonal basis for a vector space is a set of basis vectors that are mutually perpendicular with respect to the given inner product. In an inner product space, the orthogonality allows for simplified computations such as projections and decompositions since the inner product between different basis vectors is zero. Demonstrating that a set of vectors is orthogonal involves checking that the inner product of each pair of distinct vectors equals zero.

Transcript

00:02 We're given a vector space m, which is the set of all two by two matrices equipped with the inner product of a and b is the trace of b transpose times a.

00:21 And we're asked to find an orthogonal basis for the orthogonal complement of different subspaces.

00:31 So in part a, we're asked to find an orthogonal basis for the orthogonal complement of diagonal matrices.

00:47 Well, the set of, i guess i'd call them matrices a1, which is 1 -0 -0, and a2, which is 0 -0 ,000, and a2, which is 0 -01.

01:13 This is a basis for the diagonal matrices.

01:26 So to find the orthogonal matrixes, so to find the orthogonal complement, well, we want to find vectors, or sorry not vectors, but matrices, b with entries a, b, c, d, such that the inner product of b with a1 and the inner product of b with a2 are both zero.

02:12 Of course, notice that we could also write our inner product as the sum from i equals 1 to 2, some from i equals j equals 1 to 2 of a i j b i j and these two equations they imply that a equals 0 and that d equals 0 we don't have any restrictions on b and c though.

03:09 So for example, we could take b equal to 1 and c equal to 0, and so we get the matrix, i'll call it b1, which has entries 0 -1 -0.

03:31 Now we want to find an orthogonal basis for the orthogonal complement, so we want to find a matrix b2, such as a that the inner product of b2, a1, the interproduct of b2, a2, and the inner product of b2, and b2, and b1 are all zero.

03:54 And therefore we get the same two equations, a equals 0, d equals 0.

03:58 But then we also get this equation, b equals 0.

04:11 0.

04:18 So we have one degree of freedom.

04:19 C, well, let's just say c equals 1.

04:22 And we obtain another matrix b2, which is 0 -010.

04:32 And so it follows that the set of matrices b1, b2 is an orthogonal basis for the orthogonal complement to the set of matrices b1, b2 is an orthogonal basis for the orthogonal complement to the set of diagonal matrices...

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