Call Now to Set Up Tutoring:

(888) 888-0446

Linear Algebra : Linear Independence and Rank

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 #1 : Linear Independence And Rank

Determine whether the following vectors in Matrix form are Linearly Independent.

$A=\begin{bmatrix} 2 & 4 & 10\\ 3 & -7 & 11\\ -1 & 4 & 10 \end{bmatrix}$

Possible Answers:

The vectors are Linearly Independent

The vectors aren't Linearly Independent

Correct answer:

The vectors are Linearly Independent

Explanation:

To figure out if the matrix is independent, we need to get the matrix into reduced echelon form. If we get the Identity Matrix, then the matrix is Linearly Independent.

$\begin{bmatrix} 2 & 4 & 10\\ 3 & -7 & 11\\ -1 & 4 & 10 \end{bmatrix}$

$\frac{2}{3}R_2-R_1\rightarrow R_2$

$\begin{bmatrix} 2 & 4 & 10\\ 0 & -\frac{26}{3} & -\frac{8}{3}\\ -1 & 4 & 10 \end{bmatrix}$

$-2 R_3+R_1\rightarrow R_3$

$\begin{bmatrix} 2 & 4 & 10\\ 0 & -\frac{26}{3} & -\frac{8}{3}\\ 0 & -4 & -10 \end{bmatrix}$

$\\-3R_2\rightarrow R_2 \\-R_3\rightarrow R_3$

$\begin{bmatrix} 2 & 4 & 10\\ 0 & 26 & 8\\ 0 & 4 & 10 \end{bmatrix}$

$\frac{13}{2} R_3-R_2\rightarrow R_3$

$\begin{bmatrix} 2 & 4 & 10\\ 0 & 26 & 8\\ 0 & 0 & 57 \end{bmatrix}$

$\frac{13}{2} R_3-R_2\rightarrow R_3$

$\begin{bmatrix} 2 & 4 & 10\\ 0 & 26 & 8\\ 0 & 0 & 57 \end{bmatrix}$

$\\\frac{1}{2}R_1\rightarrow R_1 \\ \\ \frac{1}{2}R_2\rightarrow R_2$

$\begin{bmatrix} 1 & 2 & 5\\ 0 & 13 & 4\\ 0 & 0 & 57 \end{bmatrix}$

$\frac{4}{57} R_3-R_2\rightarrow R_2$

$\begin{bmatrix} 1 & 2 & 5\\ 0 & -9 & 0\\ 0 & 0 & 57 \end{bmatrix}$

$\frac{5}{57} R_3-R_1\rightarrow R_1$

$\begin{bmatrix} 4 & 3 & 0\\ 0 & -9 & 0\\ 0 & 0 & 57 \end{bmatrix}$

$R_2+3R_1\rightarrow R_1$

$\begin{bmatrix} 12 & 0 & 0\\ 0 & -9 & 0\\ 0 & 0 & 57 \end{bmatrix}$

$\\\frac{1}{12}R_1\rightarrow R_1 \\ \\ -\frac{1}{9}R_2\rightarrow R_2 \\ \\ \frac{1}{57}R_3\rightarrow R_3$

$\begin{bmatrix} 1 & 0 & 0\\ 0 & 1& 0\\ 0 & 0 & 1 \end{bmatrix}$

Since we got the Identity Matrix, we know that the matrix is Linearly Independent.

Report an Error

Example Question #2 : Linear Independence And Rank

Find the rank of the following matrix.

$\begin{bmatrix} 10 & 2 \\ 4 & 5\\ 5& 10 \end{bmatrix}$

Possible Answers:

Rank=0

Rank=1

Rank=4

Rank=3

Rank=2

Correct answer:

Rank=2

Explanation:

We need to get the matrix into reduced echelon form, and then count all the non all zero rows.

$R_2-\frac{2}{5}R_1\rightarrow R_2$

$\begin{bmatrix} 10 & 2 \\ 0 & \frac{21}{5}\\ 5& 10 \end{bmatrix}$

$R_3-\frac{1}{2}R_1\rightarrow R_3$

$\begin{bmatrix} 10 & 2 \\ 0 & \frac{21}{5}\\ 0& 9 \end{bmatrix}$

$R_2 \leftrightarrow R_3$

$\begin{bmatrix} 10 & 2 \\ 0& 9 \\ 0 & \frac{21}{5}\end{bmatrix}$

$R_3-\frac{7}{15}R_2\rightarrow R_3$

$\begin{bmatrix} 10 & 2 \\ 0& 9 \\ 0 & 0\end{bmatrix}$

$\frac{1}{9}R_2\rightarrow R_2$

$\begin{bmatrix} 10 & 2 \\ 0& 1 \\ 0 & 0\end{bmatrix}$

$R_1-2R_2\rightarrow R_1$

$\begin{bmatrix} 10 & 0 \\ 0& 1 \\ 0 & 0\end{bmatrix}$

$\frac{1}{10}R_1\rightarrow R_1$

$\begin{bmatrix} 1 & 0 \\ 0& 1 \\ 0 & 0\end{bmatrix}$

The rank is 2, since there are 2 non all zero rows.

Report an Error

Example Question #3 : Linear Independence And Rank

Calculate the Rank of the following matrix

$\begin{bmatrix} 1&2 &3 \\ 1& 2& 3\\ 1&2 &3 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

We need to put the matrix into reduced echelon form, and then count all the non-zero rows.

$\\R_1-R_2\rightarrow R_2 \\R_1-R_3\rightarrow R_3$

$\begin{bmatrix} 1&2 &3 \\ 0& 0& 0\\ 0&0 &0 \end{bmatrix}$

Since there is only 1 non-zero row, the Rank is 1.

Report an Error

Example Question #1 : Linear Independence And Rank

Determine if the following matrix is linearly independent or not.

$A=\begin{bmatrix} 5 & 3\\ 10 & 6 \end{bmatrix}$

Possible Answers:

Linearly Dependent

Linearly Independent

Correct answer:

Linearly Dependent

Explanation:

Since the matrix is $2\times2$ , we can simply take the determinant. If the determinant is not equal to zero, it's linearly independent. Otherwise it's linearly dependent.

$\det(A)=5(6)-(10)(3)=30-30=0$

Since the determinant is zero, the matrix is linearly dependent.

Report an Error

Example Question #1 : Linear Independence And Rank

If matrix A is a 5x8 matrix with a two-dimensional null space, what is the rank of A?

Possible Answers:

rank( A) = 6

rank (A) = 5

None of the other answers.

rank (A) = 3

rank (A) = 8

Correct answer:

rank( A) = 6

Explanation:

Given that rank A + dimensional null space of A = total number of columns, we can determine rank A = total number of columns-dimensional null space of A. Using the information given in the question we can solve for rank A:

rank (A) = 8-2 = 6

Report an Error

Example Question #1 : Linear Independence And Rank

If matrix A is a 10x12 matrix with a three-dimensional null space, what is the rank of A?

Possible Answers:

rank( A) = 10

rank (A) = 9

rank (A) = 12

None of the other answers

rank (A) = 3

Correct answer:

rank (A) = 9

Explanation:

rank (A) = 12-3 = 9

Report an Error

Example Question #4 : Linear Independence And Rank

Does the following row reduced echelon form of a matrix represent a linearly independent set?

$\begin{bmatrix} 1 & 0& 0& 0& \\0 &0 &1 &0 & \end{bmatrix}$

Possible Answers:

Not enough information

Yes

Correct answer:

Yes

Explanation:

The set must be linearly independent because there are no rows of all zeros. There are columns of all zeros, but columns do not tell us if the set is linearly independent or not.

Report an Error

Example Question #2 : Linear Independence And Rank

In a vector space of dimension 5, can you have a linearly independent set of 3 vectors?

Possible Answers:

Not enough information

Yes

Correct answer:

Yes

Explanation:

The dimension of the vector space is the maximum number of vectors in a linearly independent set. It is possible to have linearly independent sets with less vectors than the dimension.

So for this example it is possible to have linear independent sets with

1 vector, or 2 vectors, or 3 vectors, all the way up to 5 vectors.

Report an Error

Example Question #1 : Linear Independence And Rank

Consider a set of 3 vectors from a 3 dimensional vector space.

Is the set linearly independent?

Possible Answers:

Not enough information

Yes

Correct answer:

Not enough information

Explanation:

It depends on what the vectors are.

For example, if

$\b{v}_1 = (1,0,0)$

$\b{v}_2 = (0,1,0)$

$\b{v}_3 = (0,0,1)$

Then the set is linearly independent.

However if the vectors were

$\b{v}_1 = (1,0,0)$

$\b{v}_2 = (0,1,0)$

$\b{v}_3 = (2,0,0)$

then the set would be linearly dependent.

Report an Error

Example Question #1 : Linear Independence And Rank

Consider a set of 3 vectors from a 2 dimensional vector space.

Is the set linearly independent?

Possible Answers:

Yes

Not enough information

Correct answer:

Explanation:

Since the dimension of the space is 2, a linearly independent set can have at most two vectors. Since the set in consideration has 3 and 3>2, the set must be linearly dependent.

Report an Error

View Linear Algebra Tutors

Liban
Certified Tutor

Emory University, Bachelor of Science, Mathematics/Economics.

View Linear Algebra Tutors

Michael
Certified Tutor

Cleveland State University, Bachelor of Science, Mathematics.

View Linear Algebra Tutors

Nafis
Certified Tutor

University of Leipzig, Germany., Bachelor of Science, Physics. University of Pittsburgh-Pittsburgh Campus, Doctor of Philosop...

All Linear Algebra Resources

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

Popular Subjects

Spanish Tutors in Philadelphia, Reading Tutors in Washington DC, Algebra Tutors in Boston, Algebra Tutors in Atlanta, GMAT Tutors in Denver, Spanish Tutors in San Diego, Computer Science Tutors in Atlanta, Biology Tutors in Houston, Math Tutors in San Francisco-Bay Area, LSAT Tutors in San Diego

Popular Courses & Classes

GRE Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in Seattle, SAT Courses & Classes in Miami, ISEE Courses & Classes in Denver, ISEE Courses & Classes in Dallas Fort Worth, SSAT Courses & Classes in Philadelphia, GMAT Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Los Angeles, SSAT Courses & Classes in Miami, ISEE Courses & Classes in Boston

Popular Test Prep

GMAT Test Prep in Chicago, ISEE Test Prep in New York City, SSAT Test Prep in Miami, SSAT Test Prep in San Diego, SAT Test Prep in Atlanta, GMAT Test Prep in Washington DC, GRE Test Prep in Los Angeles, ACT Test Prep in Washington DC, LSAT Test Prep in San Diego, SAT Test Prep in Phoenix

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 Independence and Rank

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #1 : Linear Independence And Rank

Example Question #2 : Linear Independence And Rank

Example Question #3 : Linear Independence And Rank

Example Question #1 : Linear Independence And Rank

Example Question #1 : Linear Independence And Rank

Example Question #1 : Linear Independence And Rank

Example Question #4 : Linear Independence And Rank

Example Question #2 : Linear Independence And Rank

Example Question #1 : Linear Independence And Rank

Example Question #1 : Linear Independence And Rank

All Linear Algebra Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: