Call Now to Set Up Tutoring:

(888) 888-0446

Linear Algebra : Linear Mapping

Study concepts, example questions & explanations for Linear Algebra

Create An Account

All Linear Algebra Resources

4 Diagnostic Tests 108 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #51 : Linear Mapping

$M_{2 \times 1 }$ is the set of all two-by-one matrices - that is, the set of all column matrices with two entries.

Let $X= \begin{bmatrix} \sqrt{6} & -2\sqrt{3} \\- 2\sqrt{3} & 2 \sqrt{6}\end{bmatrix}$ . Define a linear mapping $T: M_{2 \times 1 } \rightarrow M_{2 \times 1 }$ as follows:

T(A) =XA .

True or false: is one-to-one and onto.

Possible Answers:

True

False; is onto but not one-to-one

False; is neither one-to-one nor onto

False; is one-to-one but not onto

Correct answer:

False; is neither one-to-one nor onto

Explanation:

The domain and the codomain of are identical, so is one to one if and only if it is onto.

A necessary and sufficient condition for to be one-to-one is that the kernel of be $\left \{ 0 \right \}$ . In $M_{2 \times 1 }$ , the zero element is $O= \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ , and this condition states that if

T(A) = O , then A = O

Thus, we can prove that is not one-to-one - and not onto - by finding a nonzero column matrix such that T(A)= XA = O .

Set $A= \begin{bmatrix} \sqrt{2} \\ 1 \end{bmatrix}$ . Then

T(A)= XA

$= \begin{bmatrix} \sqrt{6} & -2\sqrt{3} \\- 2\sqrt{3} & 2 \sqrt{6}\end{bmatrix}\begin{bmatrix} \sqrt{2} \\ 1 \end{bmatrix}$

$= \begin{bmatrix} \sqrt{6}( \sqrt{2} )+( -2\sqrt{3}) (1)\\-2 \sqrt{3} (\sqrt{2}) +( 2\sqrt{6}) (1) \end{bmatrix}$

$= \begin{bmatrix} 2\sqrt{3} -2\sqrt{3} \\-2 \sqrt{6}+ 2\sqrt{6} \end{bmatrix}$

$=\begin{bmatrix} 0 \\ 0 \end{bmatrix} = O$

There is at least one nonzero column matrix in the kernel of , so is not one-to-one or onto.

Report an Error

Example Question #52 : Linear Mapping

is the set of all polynomials of finite degree in .

Define a linear mapping $T:P \rightarrow P$ as follows:

$T(p(x)) =\frac{\mathrm{d} ^2}{\mathrm{d} x^2}p(x)$ .

True or false: T(x) is a one-to-one and onto linear mapping.

Possible Answers:

False: is neither one-to-one nor onto.

True

False: is onto but not one-to-one.

False: is one-to-one but not onto.

Correct answer:

False: is onto but not one-to-one.

Explanation:

The domain and the codomain are both of infinite dimension, so it is possible for be one-to-one, onto, both, or neither.

is one-to-one if and only if

T(p) = T(q) implies p =q .

Let p(x)= x+ 1 and q(x)= x+ 2

Then

$T(p)= \frac{\mathrm{d} ^2}{\mathrm{d} x^2}p(x)= 0$ and $T(q)= \frac{\mathrm{d} ^2}{\mathrm{d} x^2}q(x)= 0$ .

Since

T(p) = T(q) , but $p \ne q$ , is not one-to-one.

Now let $t(x)= \sum_{n= 0}^{\infty} a_{n}x^{n}$ , where finitely many $a_{n}$ are nonzero.If

$p (x)= \sum_{n= 0}^{\infty} \frac{a_{n}}{(n+2)(n+1)}x^{n+2}$ ,

then

$T(p)= \frac{\mathrm{d} ^2}{\mathrm{d} x^2} \left ( \sum_{n= 0}^{\infty} \frac{a_{n}}{(n+2)(n+1)}x^{n+2} \right )$

$= \sum_{n= 0}^{\infty} a_{n}x^{n} = t(x)$

is therefore onto.

Report an Error

View Linear Algebra Tutors

Allison
Certified Tutor

Asbury University, Bachelor of Education, Mathematics Teacher Education.

View Linear Algebra Tutors

Michael
Certified Tutor

Cleveland State University, Bachelor of Science, Mathematics.

View Linear Algebra Tutors

Teagan
Certified Tutor

The University of Texas at Austin, Bachelor of Science, Physics.

All Linear Algebra Resources

4 Diagnostic Tests 108 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Statistics Tutors in Chicago, Chemistry Tutors in Atlanta, French Tutors in Miami, GRE Tutors in Philadelphia, Calculus Tutors in Miami, Chemistry Tutors in Houston, MCAT Tutors in Washington DC, Reading Tutors in Dallas Fort Worth, GMAT Tutors in Miami, GRE Tutors in Atlanta

Popular Courses & Classes

ACT Courses & Classes in Los Angeles, LSAT Courses & Classes in Chicago, GMAT Courses & Classes in Houston, SAT Courses & Classes in Atlanta, ACT Courses & Classes in San Francisco-Bay Area, GRE Courses & Classes in Miami, GMAT Courses & Classes in Washington DC, ACT Courses & Classes in Chicago, ACT Courses & Classes in Washington DC, GMAT Courses & Classes in Phoenix

Popular Test Prep

ACT Test Prep in San Francisco-Bay Area, MCAT Test Prep in Los Angeles, LSAT Test Prep in Washington DC, ACT Test Prep in Los Angeles, ACT Test Prep in Chicago, LSAT Test Prep in Los Angeles, SAT Test Prep in Los Angeles, GMAT Test Prep in Phoenix, SAT Test Prep in Houston, GRE Test Prep in Philadelphia

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Linear Algebra : Linear Mapping

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #51 : Linear Mapping

Example Question #52 : Linear Mapping

All Linear Algebra Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: