2020/08/01 On marking the retest
No point may be given to an answer in case it is not accompanied by calculations or explanations sufficient to justify it, even though the answer itself is correct.
- Problem 1 (1).
Most students gave a wrong answer by directly applying the formula of orthogonal projection to the given vectors.
Since the vectors are not orthogonal, one needs to find orthogonal vectors spanning the same subspace and then to apply the formula.
- Problem 2 (1). Find the representation matrix of the quadratic form.
Then find a basis of the kernel as usual.
- Problem 2 (2). Calculate the characteristic polynomial carefully.
The transition matrix Q must be orthogonal. That is, the columns of Q must be orthonormal, not just orthogonal. You do not have to calculate the inverse of Q, for the inverse of an orthogonal matrix is just the transpose.
- Problem 3 (1). Finding the minimal polynomial is easy, and you can answer the question immediately by general properties of minimal polynomials.
Many students wasted their time by calculating the characteristic polynomial of a 4 by 4 matrix.
- Problem 3 (2).
By the conditions prescribed in the question, one knows the eigenvalue, the largest size of the Jordan blocks, and the number of Jordan blocks.
- Problem 4 (1). Positivity of a symmetric matrix is determined by positivity of the leading principal minor determinants.
The relevant determinants are easily and quickly calculated by row reduction.
- Problem 4 (2). The exponential of the matrix in question is a rotation. Find the axis and the angle.