代写Linear Algebra - Fall 2023 Practice代做迭代

日期：2024-12-23 05:17

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.