4 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? |
4 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? |
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.1: Eigenvalues And Eigenvectors
Problem 78E: Find all values of the angle for which the matrix A=[cossinsincos] has real eigenvalues. Interpret...
Related questions
Question
100%
PLEASE answer number 2 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%2Fb1e9ea68-fc3d-4b6a-b3cf-3cd1130a0d68%2F0p184ye.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.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 1 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Elementary Linear Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305658004/9781305658004_smallCoverImage.gif)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![Linear Algebra: A Modern Introduction](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
![Elementary Linear Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305658004/9781305658004_smallCoverImage.gif)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![Linear Algebra: A Modern Introduction](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
![Calculus For The Life Sciences](https://www.bartleby.com/isbn_cover_images/9780321964038/9780321964038_smallCoverImage.gif)
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,