Linear Algebra Final Exam Review Problems and Solutions (a lot about Orthogonality)

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?

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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[juliachu]

This is so helpful! Thank you so much!

[HeyKevinYT]

An indispensable resource, thank you very much!

[keldonchase]

Suppose A is an orthogonal matrix.
Is the following statement true or false?
(A^-1 * x) * (A^-1 * x) = ||x||^2
The answer key says "true" and makes the argument that since A is orthogonal, A^-1 is orthogonal, and orthogonal matrices preserve the Dot Product.
However, I'm inclined to say false simply because the dimensions do not allow for proper matrix multiplication.
A^-1 must be a square matrix so suppose its dimensions are (2x2) for example.
x is a column vector so suppose its dimensions are (2x1).
||x||^2 is just a number.

Putting it all together, I find that:
((2x2)(2x1))*((2x2)(2x1)) = Number
(2x1)*(2x1) = Number
But the inner numbers (1 and 2) do not match so I don't believe this product of matrices is possible to conduct.

If we were to take the TRANSPOSE of the first, then the dimensions work out.
((2x2)(2x1))^T * ((2x2)(2x1)) = Number
(2x1)^T * (2x1) = Number
(1x2) * (2x1) = Number
This way, the dimensions work out since the inner numbers (2 and 2) match.
(1x1) = Number
Number = Number

So I would argue (A^-1 * x) * (A^-1 * x) = ||x||^2 is FALSE.
Yet (A^-1 * x)^T * (A^-1 * x) = ||x||^2 is TRUE.

I'm curious to see if you would have any insight on this.
Thank you so much!
:)

[JingwenXia]

Thank you! So helpful to understand all concepts.