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 differenteigenvalues.(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 aneigenvalue c, then v is an eigenvector for A- with respect to the eigenvalue c4(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 54. Consider the matrix4.-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].%3D5. 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 matrixthat represent the development of a population of a0.70.certain mammal that leaves 3 years. In the long term, what is the percentage of the populationin the last year of life.10. Let v = [1, 1] and w = [2, 1] be two vectors: find the project of v into w%3D11. Consider the vector space generated by [1, 1, 1, 1), [1, 1, 2, 4], [1, 2, – 4, –3]. Find an orthogonalbasis.

Question

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 differenteigenvalues.(c) If the vector v is an eingenvector for the eigenvalue = ; show that lim, m A&#34;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 aneigenvalue c, then v is an eigenvector for A- with respect to the eigenvalue c4(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 54. Consider the matrix4.-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].%3D5. 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 matrixthat represent the development of a population of a0.70.certain mammal that leaves 3 years. In the long term, what is the percentage of the populationin the last year of life.10. Let v = [1, 1] and w = [2, 1] be two vectors: find the project of v into w%3D11. Consider the vector space generated by [1, 1, 1, 1), [1, 1, 2, 4], [1, 2, – 4, –3]. Find an orthogonalbasis.

Accepted Answer

Given vectors are , 
[ 1,2,0,-1 ] ,  [ 2,6,-3,-3 ] ,  [ 3 ,10 ,-6 ,-5 ] 
(.)  The given vectors are…

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? |