We are open Saturday and Sunday!

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

← Previous 1 2 3 4 5 6 Next →

Example Question #31 : Linear Mapping

A linear map $f:R^4 \rightarrow R^3$ has a rank of 3. Is the linear map, , 1-to-1?

(Hint- Use the formula d=n+r where

is the dimension of the domain

is the dimension of the null space

is the rank of the linear map )

Possible Answers:

Not enough information

Yes

Correct answer:

Explanation:

The answer is no because the dimension of the null space is not zero. This comes from the equation

d=n+r

We know that the domain is R^4 which has dimension of . Therefore

d=4

Also from the problem statement r=2 .

Plugging these into the equation gives

d=n+r

4=n+2

n=2

Since the dimension of the null space is and not , then the function can't be 1-to-1.

Report an Error

Example Question #31 : Linear Mapping

A linear map $f:R^4 \rightarrow R^2$ has a null space consisting of only the zero vector. Is 1-to-1?

Possible Answers:

Yes

Not enough information

Correct answer:

Yes

Explanation:

If the dimension of the null space is zero then the linear map is 1-to-1.

For this problem, we are told the null space is only the zero vector. Therefore the null space has dimension . Since the null space has dimension , then is 1-to-1.

Report an Error

Example Question #31 : Linear Mapping

A linear map $f:R^4 \rightarrow R^5$ has a rank of 4. Is the linear map, , 1-to-1?

(Hint- Use the formula d=n+r where

is the dimension of the domain

is the dimension of the null space

is the rank of the linear map )

Possible Answers:

Yes

Not enough information

Correct answer:

Yes

Explanation:

The answer is yes because the dimension of the domain and the rank are equal. This implies that the dimension of the null space is zero.

This comes from the equation

d=n+r

We know that the domain is R^4 which has dimension of . Therefore

d=4

Also from the problem statement r=4 .

Plugging these into the equation gives

d=n+r

4=n+4

n=0

Since the dimension of the null space is then the function is 1-to-1.

Notice that whenever d=r , then is always zero. Thus whenever d=r , the linear mapping is 1-to-1.

Report an Error

Example Question #34 : Linear Mapping

The null space (sometimes called the kernal) of a mapping is a subspace in the domain such that all vectors in the null space map to the zero vector.

Consider the mapping $f:R^2\rightarrow R^3$ such that f(x,y) = (5x+5y,x+y,0) .

What is the the null space of ?

Possible Answers:

The space spanned by the vector (1,-1)

R^2

R^3

The zero vector

Correct answer:

The space spanned by the vector (1,-1)

Explanation:

Any vector in the null space satisfies f(x,y) = (0,0,0) .

Therefore we get the following equation:

(5x+5y,x+y,0) = (0,0,0)

Thus x=-y . Hence the null space is any vector in form (a,-a) where is any real number. Therefore, any point on the line y=x gets mapped to the zero vector in R^3

Report an Error

Example Question #35 : Linear Mapping

This problem deals with the zero map. I.e the map that takes all vectors to the zero vector.

Consider the mapping $f:R^2\rightarrow R^3$ such that f(x,y) = (0,0,0) .

What is the the null space of ?

Possible Answers:

R^2

R^3

The line y=x

The vector (0,0)

Correct answer:

R^2

Explanation:

The zero map takes all vectors to the zero vector. Therefore, the entire domain of the map is the null space. The domain of this map is R^2 . Thus R^2 is the null space.

Report an Error

Example Question #891 : Linear Algebra

This problem deals with the zero map. I.e the map the takes all vectors to the zero vector.

Consider the mapping $f:R^2\rightarrow R^3$ such that f(x,y) = (0,0,0) .

What is the the rank of ?

Possible Answers:

Correct answer:

Explanation:

The image of the the zero map is the zero vector. A single vector has dimension . Therefore the dimension of the image is zero. Hence the rank is zero.

Report an Error

Example Question #37 : Linear Mapping

$C_{ \mathbb{R}}$ refers to the set of all functions with domain $\mathbb{R}$ and range a subset of $\mathbb{R}$ .

Define the transformation $T: C_{ \mathbb{R}} \rightarrow \mathbb{R}$ to be

$T(f) = \sum_{j=1}^{100} f'(j)$

True or false: is a linear transformation.

Possible Answers:

False

True

Correct answer:

True

Explanation:

For to be a linear transformation, it must hold that

T(f+g)= T(f)+ T(g)

and

T(cf) = cT(f)

for all f,g in the domain of and and for all scalar .

Let $f, g \in C_{ \mathbb{R}}$ .

$T(f) = \sum_{j=1}^{100} f'(j)$ and $T(g) = \sum_{j=1}^{100} g'(j)$ , so

$T(f)+ T(g)= \sum_{j=1}^{100} f'(j)+ \sum_{j=1}^{100} g'(j)$

By the sum rule for finite sequences,

$T(f)+ T(g)= \sum_{j=1}^{100}[ f'(j) + g'(j)]$

By the derivative sum rule,

$T(f)+ T(g)= \sum_{j=1}^{100}( f '+g')(j)$

$T(f)+ T(g)= \sum_{j=1}^{100}( f +g)'(j)$

T(f)+ T(g)= T(f+g)

The first condition is met.

Let $f \in C_{ \mathbb{R}}$ and be a scalar.

$T(f) = \sum_{j=1}^{100} f'(j)$

$c T(f) = c\sum_{j=1}^{100} f'(j)$

By the scalar product rule for finite sequences,

$c T(f) = \sum_{j=1}^{100}c f'(j)$

By the scalar product rule for derivatives,

$c T(f) = \sum_{j=1}^{100}(c f)'(j)$

c T(f) =T(c f)

The second condition is met.

is a linear transformation.

Report an Error

Example Question #31 : Linear Mapping

$C_{[0, \pi]}$ refers to the set of all functions that are continuous on $[0, \pi]$ .

Define the linear mapping $T: C_{[0, \pi]}\rightarrow \mathbb{R}$ as follows:

$T(f)= \int_{0}^{ \pi} f(x) \textup{ d}x$

True or false: $f(x) = \cos x + \sin x$ is in the kernel of .

Possible Answers:

True

False

Correct answer:

False

Explanation:

The kernel of a linear transformation is the subset of the domain of that maps into the zero of its range, so, by definition, $f \in \ker (T)$ if and only if

$T(f)= \int_{0}^{ \pi} f(x) \textup{ d}x = 0$

To determine whether this is true or false, evaluate the integral:

$T(f)= \int_{0}^{ \pi} (\cos x + \sin x) \textup{ d}x = 0$

$=\ \left.\begin{matrix} \sin x - \cos x \\ \; \end{matrix}\right|\ \begin{matrix} \pi \\ 0 \end{matrix}$

$= (\sin \pi - \cos \pi) - (\sin 0 - \cos 0 )$

= (0 - (- 1)) - (0-1)

$T(f) \ne 0$ , so $f \notin \ker (T)$ .

Report an Error

Example Question #39 : Linear Mapping

$C_{[0, 2\pi]}$ refers to the set of all functions that are continuous on $[0,2\pi]$ .

Define the linear mapping $T: C_{[0, 2\pi]}\rightarrow \mathbb{R}$ as follows:

$T(f)= \int_{0}^{2 \pi} f(x) \textup{ d}x$

True or false: $f(x) = \cos x + \sin x$ is in the kernel of .

Possible Answers:

False

True

Correct answer:

True

Explanation:

The kernel of a linear transformation is the subset of the domain of that maps into the zero of its range. It follows that $f \in \ker (T)$ if and only if

$T(f)= \int_{0}^{2 \pi} f(x) \textup{ d}x = 0$

To determine whether this is true or false, evaluate the integral:

$T(f)= \int_{0}^{2 \pi} (\cos x + \sin x) \textup{ d}x = 0$

$=\ \left.\begin{matrix} \sin x - \cos x \\ \; \end{matrix}\right|\ \begin{matrix} 2 \pi \\ 0 \end{matrix}$

$= (\sin 2\pi - \cos 2 \pi) - (\sin 0 - \cos 0)$

= (0 - 1) - (0-1)

= 0

Therefore, $f \in \ker (T)$ .

Report an Error

Example Question #521 : Operations And Properties

True or false: If $T: V \rightarrow W$ is a linear mapping, and is a vector space, then is a subspace of .

Possible Answers:

False

True

Correct answer:

False

Explanation:

For example, if is the space of all vectors in $\mathbb{R}^2$ of the form (a,3a) , and is the space of all vectors in $\mathbb{R}^2$ the form (a,0) , then $T: (a_1,a_2) \rightarrow (a_1,0)$ is a linear mapping, but is not a subset of , let alone a subspace of .

Report an Error

← Previous 1 2 3 4 5 6 Next →

View Linear Algebra Tutors

Liban
Certified Tutor

Emory University, Bachelor of Science, Mathematics/Economics.

View Linear Algebra Tutors

Messan
Certified Tutor

Indiana University-Purdue University-Indianapolis, Bachelor of Engineering, Electrical Engineering.

View Linear Algebra Tutors

Sarah
Certified Tutor

Elizabethtown College, Bachelor of Science, Biotechnology.

All Linear Algebra Resources

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

Popular Subjects

SAT Tutors in Miami, GRE Tutors in Los Angeles, ISEE Tutors in Boston, Physics Tutors in Houston, Algebra Tutors in Boston, Computer Science Tutors in Boston, French Tutors in New York City, Calculus Tutors in Atlanta, French Tutors in Boston, Chemistry Tutors in San Francisco-Bay Area

Popular Courses & Classes

MCAT Courses & Classes in Atlanta, MCAT Courses & Classes in Chicago, Spanish Courses & Classes in New York City, SSAT Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in Philadelphia, Spanish Courses & Classes in Los Angeles, Spanish Courses & Classes in San Francisco-Bay Area, GRE Courses & Classes in Houston, SSAT Courses & Classes in Philadelphia, GRE Courses & Classes in Phoenix

Popular Test Prep

SAT Test Prep in Atlanta, LSAT Test Prep in Washington DC, SAT Test Prep in Dallas Fort Worth, SSAT Test Prep in Los Angeles, GRE Test Prep in Houston, SSAT Test Prep in Atlanta, SAT Test Prep in New York City, LSAT Test Prep in Houston, GRE Test Prep in Phoenix, LSAT 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 #31 : Linear Mapping

Example Question #31 : Linear Mapping

Example Question #31 : Linear Mapping

Example Question #34 : Linear Mapping

Example Question #35 : Linear Mapping

Example Question #891 : Linear Algebra

Example Question #37 : Linear Mapping

Example Question #31 : Linear Mapping

Example Question #39 : Linear Mapping

Example Question #521 : Operations And Properties

All Linear Algebra Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: