1) Linear difference equation (eigenvalues, eigenvectors, & diagonalization). 2) Orthogonal diagonalization (Spectral Theorem). 3) Pythagorean Theorem. 4) λ^2 is an eigenvalue of A^2 when λ is an eigenvalue of A. 5) Orthogonal complement of a subspace W is a subspace of R^n (subspace test). 6) Orthogonal projection and orthogonal complement. 7) Real normal form of a matrix with complex number eigenvalues (change of variables from a rotation and dilation). 8) Gram-Schmidt orthogonalization process and orthogonal diagonalization of a 3x3 symmetric matrix (Spectral Theorem). 9) Gram-Schmidt orthogonalization process for an inner product space and orthogonal projection.10) Nul(A) and Col(A) (versus Nul(A) and Row(A), which are orthogonal complements). 11) Spectral Theorem. 12) Orthogonal matrix. 13) Determinant of similar matrices. 14) Orthogonality and linear independence. 15) Similar matrices. 16) Orthonormal columns. 17) n x n symmetric matrix A with distinct real eigenvalues is diagonalizable. 18) Norm of a vector in relation to dot product. 19) Positive definite quadratic form? "Linear Algebra and Its Applications", by David Lay: https://amzn.to/3VnT2aq
Links and resources
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(0:00) Types of problems
(0:32) Linear system of difference equations (use eigenvalues and eigenvectors and also use diagonalization to find a matrix power)
(10:28) Spectral Theorem: Orthogonal diagonalization of a symmetric matrix
(18:53) Pythagorean Theorem in R^n (use properties of dot products)
(21:33) λ^2 is an eigenvalue of A^2 when λ is an eigenvalue of A
(24:45) Subspace Test: orthogonal complement of a subspace W is a subspace of R^n
(28:19) Orthogonal projection of a vector along a line through another vector
(33:01) Real normal form of a matrix with complex number eigenvalues (change of variables from a rotation and dilation)
(40:58) Gram-Schmidt Orthogonalization Process and Spectral Theorem: Orthogonal diagonalization of a 3x3 symmetric matrix
(50:08) Gram-Schmidt for an inner product space C[0,1]: orthogonal projection and least squares minimization
(1:02:56) Nul(A) and Col(A) (versus Nul(A) and Row(A), which are orthogonal complements)
(1:04:41) Spectral Theorem for symmetric matrices
(1:05:04) Orthogonal matrices
(1:05:31) Determinants of similar matrices
(1:05:58) Orthogonality and linear independence
(1:06:55) Similar matrices
(1:07:22) U^(T)U = I when U has orthonormal columns
(1:08:57) An nxn matrix with n distinct real eigenvalues is diagonalizable
(1:09:14) Norm of a vector x in relationship to x^T*x (dot product of x with itself)
(1:10:05) Quadratic form: positive definite, negative definite, or indefinite?
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