Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Subset - Linear Algebra - Exam, Exams of Linear Algebra

This is the Past Exam of Linear Algebra which includes Vectors, Angle, System, Factorization, Matrix Pascal, Matrix, Reflects Vectors, Line Making Angle, Same Line etc. Key important points are: Subset, Polynomials, Form, Subspace, Demonstrate, Points Inside, Circle of Radius, Centered, Vector and a Scalar, Invertible Matrix Theorem

Typology: Exams

2012/2013

Uploaded on 02/27/2013

sehgal_98
sehgal_98 🇮🇳

4.8

(4)

137 documents

1 / 7

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
MATH 205 EXAM 2 March 20, 2008
Name:
Your grade is based on correctness, completeness, and clarity on each
exercise. You may use a calculator, but no notes, books, or other students.
Good luck!
1.) (15 pts.)
a.) (10 pts.) Consider the subset of P2of all polynomials of the form p(t) = a+bt2, where
ais in Rand bis in R. Demonstrate that this subset is a subspace of P2.
b.) (5 pts.) Let Hbe the set of points inside and on a circle of radius 2 that is centered
at the origin of the xyplane. That is, H={(x, y) : x2+y24}.Use an example
(two vectors, or a vector and a scalar) to show that His not a subspace of R2.
1
pf3
pf4
pf5

Partial preview of the text

Download Subset - Linear Algebra - Exam and more Exams Linear Algebra in PDF only on Docsity!

MATH 205 EXAM 2 March 20, 2008

Name:

Your grade is based on correctness, completeness, and clarity on each exercise. You may use a calculator, but no notes, books, or other students. Good luck!

1.) (15 pts.)

a.) (10 pts.) Consider the subset of P 2 of all polynomials of the form p(t) = a + bt^2 , where a is in R and b is in R. Demonstrate that this subset is a subspace of P 2.

b.) (5 pts.) Let H be the set of points inside and on a circle of radius 2 that is centered at the origin of the xy−plane. That is, H = {(x, y) : x^2 + y^2 ≤ 4 }. Use an example (two vectors, or a vector and a scalar) to show that H is not a subspace of R^2.

2.) (15 pts.) Use the Invertible Matrix Theorem (IMT) to respond to the following questions. Be sure to state clearly which parts of the IMT you are using. In all cases, assume the matrix A is n × n.

a.) (5 pts.) If the equation Ax = 0 has a nontrivial solution, then does A have fewer than n pivot positions?

b.) (5 pts.) If 0 is not an eigenvalue of A, is AT^ invertible?

c.) (5 pts.) Suppose the columns of A form a basis for Rn. What can you say about dim Col A? What can you say about dim Nul A?

4.) (15 pts.)

a.) (7 pts.) Is λ = 2 an eigenvalue of A =

? Why or why not?

b.) (8 pts.) Is u =

[

]

an eigenvector of B =

[

]

? Why or why not?

5.) (15 pts.) Use any theorems or other results from this semester to respond to the following questions. Be sure to state clearly which theorems, parts of theorems, or other reasoning you are using.

a.) (5 pts.) Let A be an n × n matrix. If the equation Ax = b has at least one solution for each b in Rn, then is there a unique solution for each b in Rn?

b.) (5 pts.) If n×n matrices E and F have the property that EF = I, then must F E = I?

c.) (5 pts.) Let B be an m × n matrix, where m and n are not necessarily equal. Is it still true that dim Row B = dim Col B?

7.) (10 pts.) Let A =

. Complete the following tasks by hand, showing

appropriate work. Use the back of this page if you need more room.

a.) (5 pts.) Compute AB for B =

b.) (5 pts.) Use the algorithm discussed in Chapter 2 to compute A−^1.