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Linear Algebra Exam, Math 205A: Solutions for Matrix Operations and Subspaces, Exams of Linear Algebra

Bidhan Chandra Krishi ViswavidyalayaLinear Algebra

The final exam for linear algebra (math 205a) focusing on matrix operations, finding lu factorization, determinants, subspaces, and projections. Students are required to find the solutions for given systems, calculate determinants, find bases for subspaces, and understand the properties of skew-symmetric matrices and reflections.

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, December 13. 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. (14 points) Find the complete solution of the system
1 2 1 3
3 6 1 5
510−3−1
x1
x2
x3
x4
=
5
7
−7
.
2. (18 points) Let A=
2 3 4
6 12 14
4 21 21
.
(a) Find the LU factorization of A.
(b) Find the determinant of Land the determinant of U.
(c) Use your answer to (b) to find the determinant of A.
(d) Use your answer to (a) to solve A~x =
13
40
37
.
3. (18 points) Find a basis for each of the four subspaces associated with the matrix
A=
1210
2321
3432
3531
What is the factored form of Athat displays these bases?
4. (18 points) Let ~v1=
1
2
3
2
,~v2=
2
3
2
1
and ~v3=
5
4
1
−2
, and let Sbe the subspace of R4spanned
by ~v1and ~v2. Find the matrix Pthat projects vectors in R4onto S, and the matrix Rthat reflects vectors
in R4through S. Also find the projection ~p of ~v3onto S, and the reflection ~r of ~v3through S.
5. (16 points) As in project #2, let ~u =
u1
u2
u3
be a unit vector in R3and define U=
0−u3u2
u30−u1
−u2u10
.
(i) How is UTrelated to U? Matrices with this property are called skew symmetric.
(ii) What is U~u? What is ~uTU?
(iii) Explain how you can tell that ~u~uTis a projection matrix. What does it project onto?
(iv) Show that U2=~u~uT
−I.
pf2

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Final Exam, Math 205A (Linear Algebra)

This take-home exam is due by 5 PM on Friday, December 13. 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. (14 points) Find the complete solution of the system

x 1 x 2 x 3 x 4

  1. (18 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 (b) to find the determinant of A.

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

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

A =

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

  1. (18 points) Let ~v 1 =

,^ ~v 2 =

 and^ ~v 3 =

,^ and let^ S^ be the subspace of^ R^4 spanned

by ~v 1 and ~v 2. Find the matrix P that projects vectors in R^4 onto S, and the matrix R that reflects vectors in R^4 through S. Also find the projection ~p of ~v 3 onto S, and the reflection ~r of ~v 3 through S.

  1. (16 points) As in project #2, let ~u =

u 1 u 2 u 3

 (^) be a unit vector in R^3 and define U =

0 −u 3 u 2 u 3 0 −u 1 −u 2 u 1 0

(i) How is U T^ related to U? Matrices with this property are called skew symmetric.

(ii) What is U~u? What is ~uT^ U?

(iii) Explain how you can tell that ~u~uT^ is a projection matrix. What does it project onto?

(iv) Show that U 2 = ~u~uT^ − I.

  1. (12 points) As in project #2, the matrix

R = (cos θ) I + (1 − cos θ) ~u~uT^ + (sin θ) U

rotates vectors in R^3 counterclockwise by θ about the line through (0, 0 , 0) in the direction of ~u, where ~u and U are as in problem 5. Show that R is an orthogonal matrix, i.e., R−^1 = RT^. I can see two logical ways to go about this: either try to figure out what R−^1 must be, and show that RT^ is the same thing (some parts of problem 5 may be helpful here); or calculate RT^ R with the aid of problem 5 and see that you get I. If you choose the first way (which may be a lot less work), please explain your answer carefully.

  1. (14 points) Let A =

(i) Calculate the eigenvalues and the corresponding eigenvectors of A.

(ii) For each eigenvalue λ, find a basis for the column space of A − λI. What do you notice?

  1. (20 points) Let R =

(i) Explain how you can tell that R is a reflection matrix.

(ii) Let S be the subspace of R^4 that R reflects through. If ~v is in S, what should R~v be? Explain.

(iii) The answer to (ii) implies that we can find a basis for S by finding the nullspace of R − I. Explain why.

(iv) Find a basis for S using the idea in (iii).

(v) In doing (iv) you must also have found a basis for the row space of R − I. What significance does it have?

(vi) Find the eigenvalues and eigenvectors of R. Hint: this should require little or no extra calculation.

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)