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Take-Home Final Exam for Math 205A: Linear Algebra, Exams of Linear Algebra

Bidhan Chandra Krishi ViswavidyalayaLinear Algebra

This is a take-home final exam for a linear algebra course, covering topics such as matrix multiplication, lu factorization, determinants, subspaces, projection matrices, and eigenvalues/eigenvectors. Problems require students to apply various techniques learned in class to solve systems of linear equations, find lu factorizations, calculate determinants, and analyze subspaces and projection matrices.

Typology: Exams

2012/2013

Uploaded on 02/27/2013

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Final Exam, Math 205A (Linear Algebra)
This take-home exam is due by 5 PM on Friday, April 12. You may consult the textbook (or any other
book) and any class notes and handouts, but please do not discuss any details of this exam with
anyone except me! Please sign the appropriate place on the other side of this sheet and turn it in with
your exam. You may ask me questions about the exam, but I reserve the right to give unsatisfying answers.
Matrix multiplications and reduced row echelon forms may be done on MATLAB or a calculator, but please
show all other work.
1. (15 points) Find the complete solution of the system ļ£«
ļ£¬
ļ£­
1 1 1 āˆ’3
1 1 āˆ’3 1
1āˆ’3 1 1
āˆ’3 1 1 1
ļ£¶
ļ£·
ļ£øļ£«
ļ£¬
ļ£­
x1
x2
x3
x4
ļ£¶
ļ£·
ļ£ø=ļ£«
ļ£¬
ļ£­
9
1
āˆ’3
āˆ’7
ļ£¶
ļ£·
ļ£ø.
2. (27 points) Let A=ļ£«
ļ£­
1 2 1
2 8 āˆ’2
1āˆ’2 7
ļ£¶
ļ£ø.
(a) Find the LU factorization of A.
(b) Find the determinant of Land the determinant of U.
(c) Use your answer to (a) to solve A~x =ļ£«
ļ£­
7
āˆ’14
47
ļ£¶
ļ£ø.
(d) Use your answer to (b) to find the determinant of A.
(e) Find Lāˆ’1and Uāˆ’1.
(f) Use your answer to (e) to find Aāˆ’1.
3. (18 points) Find a basis for each of the four subspaces associated with the matrix
B=ļ£«
ļ£­
1213
2312
1327
ļ£¶
ļ£ø
What is the factored form of Bthat displays these bases?
4. (10 points) Your column space basis in problem 3 should consist of 2 vectors in R3. What is their cross
product? Is there anything interesting about the answer? Explain.
5. (18 points) Explain how you can tell that
P=1
6
ļ£«
ļ£¬
ļ£¬
ļ£¬
ļ£­
4 0 āˆ’2 0 āˆ’2
0 3 0 āˆ’3 0
āˆ’2 0 4 0 āˆ’2
0āˆ’3 0 3 0
āˆ’2 0 āˆ’2 0 4
ļ£¶
ļ£·
ļ£·
ļ£·
ļ£ø
is a projection matrix. Find a basis for the subspace Sof R5that Pprojects onto, and a basis for SāŠ„.
pf2

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Download Take-Home Final Exam for Math 205A: Linear Algebra and more Exams Linear Algebra in PDF only on Docsity!

Final Exam, Math 205A (Linear Algebra)

This take-home exam is due by 5 PM on Friday, April 12. You may consult the textbook (or any other book) and any class notes and handouts, but please do not discuss any details of this exam with anyone except me! Please sign the appropriate place on the other side of this sheet and turn it in with your exam. You may ask me questions about the exam, but I reserve the right to give unsatisfying answers. Matrix multiplications and reduced row echelon forms may be done on MATLAB or a calculator, but please show all other work.

  1. (15 points) Find the complete solution of the system

x 1 x 2 x 3 x 4

  1. (27 points) Let A =

(a) Find the LU factorization of A.

(b) Find the determinant of L and the determinant of U.

(c) Use your answer to (a) to solve A~x =

(d) Use your answer to (b) to find the determinant of A.

(e) Find Lāˆ’^1 and U āˆ’^1.

(f) Use your answer to (e) to find Aāˆ’^1.

  1. (18 points) Find a basis for each of the four subspaces associated with the matrix

B =

What is the factored form of B that displays these bases?

  1. (10 points) Your column space basis in problem 3 should consist of 2 vectors in R^3. What is their cross product? Is there anything interesting about the answer? Explain.
  2. (18 points) Explain how you can tell that

P =

is a projection matrix. Find a basis for the subspace S of R^5 that P projects onto, and a basis for SāŠ„.

  1. (20 points) Let C =

(i) Find the eigenvalues Ī» 1 and Ī» 2 of C, and the corresponding eigenvectors ~v 1 and ~v 2.

(ii) Let Ī› =

Ī» 1 0 0 Ī» 2

and let E be the matrix whose columns are ~v 1 and ~v 2. Check that CE = EĪ›.

(iii) Find Eāˆ’^1.

(iv) Use Ī›, E and Eāˆ’^1 to obtain a formula for Ck, for a generic power k. For which powers is the formula valid?

  1. (22 points) Suppose S is an s-dimensional subspace of Rn, T is a t-dimensional subspace of Rn, and every vector in S is perpendicular to every vector in T. Let Ps be the matrix that projects vectors in Rn^ onto S, and let Pt be the matrix that projects vectors in Rn^ onto T. Finally, let U be the set of all vectors in Rn which are linear combinations of the vectors in S and the vectors in T.

(i) What is the dimension of U? Explain.

(ii) Explain why Ps + Pt is symmetric.

(iii) What is Ps Pt? (Hint: what does Ps do to a vector in T ?) What is Pt Ps?

(iv) Simplify (Ps + Pt)^2 as much as you can. What do you conclude?

(v) What is the trace of Ps + Pt? How do you know?

(vi) Explain why the column space of Ps + Pt must be contained in U.

(vii) Explain why all this implies that Ps + Pt is the projection matrix onto U.

I affirm that I did not receive help from another person in doing this exam, nor did I give help to another student in the class.

(signed)