1. Mark as TRUE or FALSE. If they are true give an explanation, if they are false, give a coun- terexample. You really need to understand the reasons behind your statements. (a) Every square matrix has an eigenvalue over the real numbers. (b) Let A be an n x n matrix. A vector v can be an eigenvector with respect to two different eigenvalues. (c) If the vector v is an eingenvector for the eigenvalue A = show that limn A"v = 0, where 0 is the zero vector 0,0,..., 0]. (d) Let A be an n x n invertible matrix. A vector v can be an eigenvector with respect to an %3D eigenvalue c, then v is an eigenvector for A- with respect to the eigenvalue c-1. (e) 5 vectors in R' are always linearly dependent. (f) 4 vectors in R are always linearly independent.
1. Mark as TRUE or FALSE. If they are true give an explanation, if they are false, give a coun- terexample. You really need to understand the reasons behind your statements. (a) Every square matrix has an eigenvalue over the real numbers. (b) Let A be an n x n matrix. A vector v can be an eigenvector with respect to two different eigenvalues. (c) If the vector v is an eingenvector for the eigenvalue A = show that limn A"v = 0, where 0 is the zero vector 0,0,..., 0]. (d) Let A be an n x n invertible matrix. A vector v can be an eigenvector with respect to an %3D eigenvalue c, then v is an eigenvector for A- with respect to the eigenvalue c-1. (e) 5 vectors in R' are always linearly dependent. (f) 4 vectors in R are always linearly independent.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
PLEASE answer number 1 only, be clear and show all steps, thanks
![1 Mark as TRUE or FALSE. If they are true give an explanation, if they are false, give a coun-
terexample. You really need to understand the reasons behind your statements.
(a) Every square matrix has an eigenvalue over the real numbers.
(b) Let A be an n x n matrix. A vector v can be an eigenvector with respect to two different
eigenvalues.
(c) If the vector v is an eingenvector for the eigenvalue = ; show that lim, m A"v = 0,
where 0 is the zero vector 0,0, .. ., 0].
(d) Let A be an n x n invertible matrix. A vector v can be an eigenvector with respect to an
eigenvalue c, then v is an eigenvector for A- with respect to the eigenvalue c
4
(e) 5 vectors in R are always linearly dependent.
(f) 4 vectors in R are always linearly independent.
2. Decide whether the vectors (1, 2,0, -1], [2, 6, -3, -3], [3, 10, -6,-5] are linearly independent.
What is the dimension of the span they generate?
3. Find a basis among the vectors (1, 2, 3], [2,6, 10], [0, –3, -6], [-1, -3, -5] (which are surely lin-
early dependent as they are 4 vectors in R3).
-3 5
4. Consider the matrix
4.
-2
(a) Find the eigenvalues of A.
(b) Find two eigenvectors v and w of A with respect of the two different eigenvalues.
(c) Find the matrix P and P-l such that PlAP is diagonal.
(d) Show that v and w are linearly independent.
(e) Without using Matlab, compute A100z, where z = [4, 1].
%3D
5. Compute the angle between the vectors [1, –1, 0, 1] and [0, 1,0, 2].
6. Compute the projections of [1, –1,0, 1] on the vector [0, 1,0, 2].
7. Consider the subset S = {[x, y, z] | x² – y + z = 0}. Decide if it is a vector subspace. If it is,
-
find a basis.
8. Consider the subset S = {[x, y, z] | x – 3y - z = 0}. Decide if it is a vector subspace. If it is,
find a basis.
2.
4.
9. Consider the Leslie matrix
that represent the development of a population of a
0.7
0.
certain mammal that leaves 3 years. In the long term, what is the percentage of the population
in the last year of life.
10. Let v = [1, 1] and w = [2, 1] be two vectors: find the project of v into w
%3D
11. Consider the vector space generated by [1, 1, 1, 1), [1, 1, 2, 4], [1, 2, – 4, –3]. Find an orthogonal
basis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b85ccec-62e5-4bbc-baaa-2833680f80af%2F4ca79fd7-2cd2-447f-ab68-6717dc78a776%2F6xqv4r9.jpeg&w=3840&q=75)
Transcribed Image Text:1 Mark as TRUE or FALSE. If they are true give an explanation, if they are false, give a coun-
terexample. You really need to understand the reasons behind your statements.
(a) Every square matrix has an eigenvalue over the real numbers.
(b) Let A be an n x n matrix. A vector v can be an eigenvector with respect to two different
eigenvalues.
(c) If the vector v is an eingenvector for the eigenvalue = ; show that lim, m A"v = 0,
where 0 is the zero vector 0,0, .. ., 0].
(d) Let A be an n x n invertible matrix. A vector v can be an eigenvector with respect to an
eigenvalue c, then v is an eigenvector for A- with respect to the eigenvalue c
4
(e) 5 vectors in R are always linearly dependent.
(f) 4 vectors in R are always linearly independent.
2. Decide whether the vectors (1, 2,0, -1], [2, 6, -3, -3], [3, 10, -6,-5] are linearly independent.
What is the dimension of the span they generate?
3. Find a basis among the vectors (1, 2, 3], [2,6, 10], [0, –3, -6], [-1, -3, -5] (which are surely lin-
early dependent as they are 4 vectors in R3).
-3 5
4. Consider the matrix
4.
-2
(a) Find the eigenvalues of A.
(b) Find two eigenvectors v and w of A with respect of the two different eigenvalues.
(c) Find the matrix P and P-l such that PlAP is diagonal.
(d) Show that v and w are linearly independent.
(e) Without using Matlab, compute A100z, where z = [4, 1].
%3D
5. Compute the angle between the vectors [1, –1, 0, 1] and [0, 1,0, 2].
6. Compute the projections of [1, –1,0, 1] on the vector [0, 1,0, 2].
7. Consider the subset S = {[x, y, z] | x² – y + z = 0}. Decide if it is a vector subspace. If it is,
-
find a basis.
8. Consider the subset S = {[x, y, z] | x – 3y - z = 0}. Decide if it is a vector subspace. If it is,
find a basis.
2.
4.
9. Consider the Leslie matrix
that represent the development of a population of a
0.7
0.
certain mammal that leaves 3 years. In the long term, what is the percentage of the population
in the last year of life.
10. Let v = [1, 1] and w = [2, 1] be two vectors: find the project of v into w
%3D
11. Consider the vector space generated by [1, 1, 1, 1), [1, 1, 2, 4], [1, 2, – 4, –3]. Find an orthogonal
basis.
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