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f23 math 307 quiz 12 problem 05 We compute the determinant of a 4x4 matrix by using the permutation characterization of the determinant.

f23 math 307 quiz 12 problem 04 We compute the determinant of a 2x2-matrix by finding the product of its eigenvalues.

f23 math 307 quiz 12 problem 03 We prove that changing the basis with respect to which the matrix of operator is written does not change the trace of the matrix.

f23 math 307 quiz 12 problem 02 We compute an unknown diagonal entry of a matrix whose eigenvalues are known.

f23 math 307 quiz 12 problem 01 We prove a couple of properties of the Riesz element corresponding to a linear functional (guaranteed to exist by the Riesz representation theorem).

f23 math 307 quiz 11 problem 03 04 We state the complex spectral theorem and then apply it to show that an operator that is unitarily equivalent to a diagonalizable operator is normal, i.e., the vector space has an orthonormal eigenbasis with respect to that operator.

f23 math 307 quiz 11 problem 02 We define what it means for an operator to be normal and then show that an operator, whose matrix with respect to an orthonormal basis is given, is normal and not self-adjoint.

f23 math 307 quiz 11 problem 01 Given bases of the same length for two finite-dimensional vector spaces, we use the universal property of a basis to define an explicit isomorphism between the two spaces.

f23 math 307 We define the trace of a matrix, show that is is basis-independent and then use this to define the trace of an operator. We establish some useful properties of the trace and then apply it in a couple of situations. 0:00, Intro 0:30, Definition of the trace of a matrix 1:14, Example of the trace 1:45, Remark on basis invariance 2:38, Theorem: Tr(AB) = Tr(BA) 6:45, New notation for t...

f23 math 307 quiz 10 problem 06 We prove an important property of self-adjoint operators, namely that inner product of a vector with its image is real.

f23 math 307 quiz 10 problem 05 We recall how to determine whether an operator is self-adjoint by finding the conjugate transpose of its matrix (with respect to an orthonormal basis, of course).

f23 math 307 quiz 10 problem 04 We prove the commonly property that the adjoint of a composition is the reverse of the composition of the adjoints.

f23 math 307 quiz 10 problem 03 Given a formula for an operator from ℝ² to ℝ⁴ and element of ℝ⁴, we find an explicit value for the adjoint applied to that given element.

f23 math 307 quiz 10 problem 02 In response to several student errors, we give an example of an invertible diagonal matrix whose diagonal entries are all the same-the identity matrix!

Finding the matrix of the operator given an eigenvalue

Explicitly finding the orthogonal complement

Orthogonal complements reverse subset containment

Proof of the Existence of Riesz elements

Applying the (modified) Gram-Schmidt method

Upper-triangular with respect to an orthonormal basis

Finding eigenvalues and eigenvectors at the same time

Pulling a scalar out of a norm

Orthogonality of the orthogonal decomposition

Verifying the inner-product axioms

The transpose is a linear isomorphism

Enough distinct eigenvalues yields an eigenbasis

The significance of a diagonal matrix

Properties of upper triangular matrices

Range of a composition