Linear Algebra - Fall 2023
Practice
1. Calculations of determinants and interpretations.
Section 6.1 # 19,21,31,34
Section 6.2 # 5,9,17,25
Section 6.3 # 3,4
Miscellaneous problems to know and really prepare your understanding
2. Show that any projection is diagonalizable.
What are the eigenvalues of a projection, and the rank in terms of its eigenvalues? (Hint: consider the number of non-zero eigenvalues).
3. Show that any reflection is diagonalizable:
Let V be a subspace of R
n
, with an orthonormal basis v1, . . . , vk, and let Q be the matrix consisting of these vectors as columns. Express the reflection R about V in terms of the matrix Q (first, recall how to express the reflection in terms of the projection onto V ). Show that R is diagonalizable, and determine the eigenvalues of R.
4. The projection of R
5 onto a subspace V has rank 3. What is the dimension of V ?
5. Consider the matrix
Determine if the matrix is diagonalizable, and, if so, find an invertible matrix S and diagonal matrix D such that S
−1AS = D.
6. Let T : R
4 → R
4 be defined by
(a) Show that T is linear.
(b) Find the matrix A of T with respect to the standard basis e1, e2, e3, e4.
(c) Diagonalize A and find an invertible matrix S and adiagonal D such that S
−1AS = D. (see above)
(d) Possibly using (c), find a basis in which the matrix of T is diagonal.
7. Is the matrix in the previous two exercises orthogonally diagonalizable? Explain and if so, find an orthogonal matrix S and diagonal matrix D such that S
TAS = D.
8. Answer the same two questions as above for the matrix:
Equivalentlty, same questions for the linear transformation T(x) = Ax.
9. Let
(a) Find an invertible matrix S and a diagonal matrix D such that S
−1AS = D.
(b) Possibly using, find a formula for An
, n ≥ 1.
10. A 4 × 4 matrix has characteristic polynomial λ
4 − λ
2
. Suppose rank(A) = 2. Show that A is diagonalizable and find its diagonal form. (a diagonal matrix D equivalent to A).
11. A 3 × 3 matrix A has characteristic polynomial −λ
3 + 4λ. Is A diagonalizable? If so find its diagonal form.
12. Let A be a 3 × 3 matrix which has λ = 0 as an eigenvalue of algebraic multiplicity 3. Calculate det(A − I)
3
.
13. A diagonalizable 5 × 5 matrix A has characteristic polynomial f(λ) = −λ
3
(λ − 1)2
. What is the rank of A − I? How about the rank of A?
14. [Comprehensive problem]
Consider the plane V given by the equation x + y + z = 0.
(a) Find a matrix whose kernel is V .
(b) Find a basis in V .
(c) Find an orthonormal basis in V .
(d) Let P be the projection onto V . Find the matrix of P.
(e) Find a basis where the matrix of P is diagonal, and determine this diagonal matrix.
(f) Find the eigenvalues of P, the determinant of P and the trace of P.
15. Let P1 = {a+bx|a, b ∈ R} be the vector space of polynomials of degree at most 1. Let T : P1 → P1 be defined by T(f) = 3f(x) − 2f(−1) + 4f(0)x.
(a) Check that T is a linear function.
(b) Find the matrix A of T with respect to the basis (1, x) of P1.
(c) Show that A is diagonalizable and find an invertible matrix S and diagonal matrix D such that S−1AS = D.
(d) Possibly using S as a change of basis matrix, find a basis B of P1 such that the matrix of T with respect to this basis is diagonal.