Module 8 Quiz: Applications of Linear Algebra
Quiz
10 questions on change of basis, SVD, PCA, and computer graphics.
1
What does the change-of-basis matrix P_B do?
P_B converts B-coordinates to standard coordinates: x = P_B * [x]_B. Its inverse converts the other way.
2
Two matrices A and B are similar. What properties must they share?
Eigenvalues, determinant, trace, rank, and characteristic polynomial.
3
In the SVD A = U Sigma V^T, what are the singular values?
The square roots of the eigenvalues of A^T A, arranged in decreasing order on the diagonal of Sigma.
4
The SVD exists for what types of matrices?
Every matrix -- square or rectangular, invertible or singular. This is a key advantage over diagonalization.
5
In PCA, what are the principal components?
The eigenvectors of the covariance matrix, ordered by eigenvalue (largest first). They are the directions of maximum variance.
6
Eigenvalues of a covariance matrix: 8, 4, 2. How much variance does PC1 + PC2 explain?
(8+4)/(8+4+2) = 12/14 = 85.7%.
7
Why do computer graphics use homogeneous coordinates?
To represent translation as matrix multiplication. Translation is not a linear map in standard coordinates, but it becomes one in homogeneous coordinates.
8
Write the 2x2 rotation matrix for 45 degrees.
R(45) = [cos(45) -sin(45); sin(45) cos(45)] = [sqrt(2)/2, -sqrt(2)/2; sqrt(2)/2, sqrt(2)/2].
9
To apply T1 then T2, what is the combined matrix?
M = T2 * T1. The rightmost matrix is applied first.
10
A matrix has singular values 10, 5, 2, 0.1. What rank approximation captures most information?
Rank 2 captures (100+25)/(100+25+4+0.01) = 125/129.01 = 96.9% of the total energy (Frobenius norm squared).