We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Linear Algebra : Range and Null Space of a Matrix

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 Next →

Example Question #11 : Range And Null Space Of A Matrix

What is the largest possible rank of a $7 \times 2$ matrix?

Possible Answers:

None of the other answers

Correct answer:

Explanation:

The rank is equal to the dimension of the row space and the column space (both spaces always have the same dimension). This matrix has seven rows and two columns, which means the largest possible number of vectors in a basis for the column space of a $7 \times 2$ matrix is , so this is the largest possible rank.

Report an Error

Example Question #11 : Range And Null Space Of A Matrix

What is the smallest possible nullity of a $7 \times 2$ matrix?

Possible Answers:

None of the other answers

Correct answer:

Explanation:

According to the Rank + Nullity Theorem,

Rank(A)+Nullity(A)= columns

Since the matrix has columns, we can rearrange the equation to get

Nullity(A) = 2- Rank(A)

So to make the nullity as small as possible, we need to make the rank as large as possible.

The rank is equal to the dimension of the row space and the column space (both spaces always have the same dimension). This matrix has three rows and five columns, which means the largest possible number of vectors in a basis for the row space of a $7 \times 2$ matrix is , so this is the largest possible rank.

Hence the smallest possible nullity is 2-(2) =0 .

Report an Error

Example Question #383 : Linear Algebra

A matrix with five rows and four columns has rank 3.

What is the nullity of ?

Possible Answers:

Correct answer:

Explanation:

The sum of the rank and the nullity of any matrix is always equal to to the number of columns in the matrix. Therefore, a matrix with four columns and rank 3, such as , must have as its nullity 4- 3= 1 .

Report an Error

Example Question #11 : Range And Null Space Of A Matrix

$C(-\infty , \infty)$ , the set of all continuous functions defined on $(-\infty, \infty)$ , is a vector space under the usual rules of addition and scalar multiplication.

True or false: The set of all functions of the form

f(x) = 2nx + n ,

where is a real number, is a subspace of $C(-\infty , \infty)$ .

Possible Answers:

True

False

Correct answer:

True

Explanation:

A subset of a vector space is a subspace of that vector space if and only if it meets two criteria. Both will be given and tested, letting

$S = \left \{ f(x)| f(x) = 2n x + n , n \in \mathbb{R}\right \}$ .

This can be rewritten as

$S = \left \{ f(x)| f(x) = n( 2 x +1) , n \in \mathbb{R}\right \}$

One criterion for to be a subspace is closure under addition; that is:

If $a, b \in S$ , then $a+b \in S$ .

Let $f, g \in S$ as defined. Then for some real n, m :

f(x) = n( 2 x +1)

g(x) = m( 2 x +1)

(f+g)(x) = f(x) + g(x)

= n( 2 x +1) + m( 2 x +1)

= (n + m)( 2 x +1)

It follows that $f+g \in S$ .

The second criterion for to be a subspace is closure under scalar multiplication; that is:

If $a \in S , c \in \mathbb{R}$ , then $ca \in S$

Let $h \in S$ as defined. Then for some real :

h(x) =p (2 x +1)

(ch)(x) = c (h(x))

$= c \cdot p (2 x +1)$

=( c p )(2 x +1)

It follows that $ch \in S$

, as defined, is a subspace of $C(-\infty , \infty)$ .

Report an Error

Example Question #385 : Linear Algebra

$C(-\infty , \infty)$ , the set of all continuous functions defined on $(-\infty, \infty)$ , is a vector space under the usual rules of addition and scalar multiplication.

True or false: The set of all functions of the form

f(x) = 3nx + n + 1 ,

where is a real number, is a subspace of $C(-\infty , \infty)$ .

Possible Answers:

True

False

Correct answer:

False

Explanation:

A subset of a vector space can be proved to not be a subspace of the space by showing that the zero of the space is not in .

Let be the subset in question, and let f(x) = 0 , the zero function, which is in $C(-\infty, \infty)$ . This cannot be expressed as f(x) = 3nx + n + 1 for any . It if could then

3n = 0 , in which case n = 0 ,

and

n+ 1 = 0 , in which case n = -1 .

By contradiction, $0 \notin S$ . is not a subspace of $C(-\infty, \infty)$ .

Report an Error

Example Question #386 : Linear Algebra

$C(-\infty , \infty)$ , the set of all continuous functions defined on $(-\infty, \infty)$ , is a vector space under the usual rules of addition and scalar multiplication.

True or false: The set of all functions of the form

f(x)= |nx |

where is a real number, is a subspace of $C(-\infty , \infty)$ .

Possible Answers:

False

True

Correct answer:

False

Explanation:

Let $S = \left \{ f(x)| f(x) = |nx | , n \in \mathbb{R}\right \}$ .

We can show through counterexample that this is not a subspace of $C(-\infty , \infty)$ .

Let f(x) = |3x| . This is an element of .

One condition for to be a subspace of a vector space is closure under scalar multiplication. Multiply f(x) by scalar . The product is the function

g(x) = -1 f(x) = -|3x| .

$g(x) \ne S$ .

This violates a criterion for a subspace, so is not a subspace of $C(-\infty , \infty)$ .

Report an Error

Example Question #11 : Range And Null Space Of A Matrix

$C(-\infty , \infty)$ , the set of all continuous functions defined on $(-\infty, \infty)$ , is a vector space under the usual rules of addition and scalar multiplication.

True or false: The set of all functions defined on $(-\infty, \infty)$ with inverses is a subspace of $C(-\infty , \infty)$ .

Possible Answers:

False

True

Correct answer:

False

Explanation:

Let be the set of all functions on $(-\infty, \infty)$ such that $f^{-1}$ is defined.

A sufficient condition for to not be a subspace of $C (-\infty, \infty)$ is that the zero of the set - which here is the zero function f(x) = 0 - is not in . $f \notin S$ because the zero function - a constant function - does not have an inverse ( f(0) = f(1) = 0 , violating a condition of an invertible function). It follows that is not a subspace of $C (-\infty, \infty)$ .

Report an Error

Example Question #388 : Linear Algebra

$C(-\infty , \infty)$ , the set of all continuous real-valued functions defined on $(-\infty, \infty)$ , is a vector space under the usual rules of addition and scalar multiplication.

Let be the set of all functions of the form

$f(x) = \left\{\begin{matrix} 0,& x< 0\\ nx,& x \ge 0 \end{matrix}\right.$

True or false: is a subspace of $C(-\infty , \infty)$ .

Possible Answers:

False

True

Correct answer:

True

Explanation:

A set is a subspace of a vector space if and only if two conditions hold, both of which are tested here.

The first condition is closure under addition - that is:

If $a , b \in S$ , then $a+ b \in S$

Let $f , g \in S$ as defined. Then for some $n, m \in \mathbb{R}$ ,

$f(x) = \left\{\begin{matrix} 0,& x< 0\\ nx,& x \ge 0 \end{matrix}\right.$

and

$g(x) = \left\{\begin{matrix} 0,& x< 0\\mx,& x \ge 0 \end{matrix}\right.$ .

$(f+ g )(x) = \left\{\begin{matrix} 0+ 0,& x< 0\\nx + mx,& x \ge 0 \end{matrix}\right.$

$(f+ g )(x) = \left\{\begin{matrix} 0 ,& x< 0\\(n + m)x,& x \ge 0 \end{matrix}\right.$ ,

$f +g \in S$ . The first condition is met.

The second condition is closure under scalar multiplication - that is:

If $a \in S$ and is a scalar, then $ca \in S$

Let $h \in S$ as defined. Then for some $p \in \mathbb{R}$ ,

$h(x) = \left\{\begin{matrix} 0,& x< 0\\ px,& x \ge 0 \end{matrix}\right.$

For any scalar ,

$c \cdot h(x) = \left\{\begin{matrix} c \cdot 0,& x< 0\\ c \cdot px,& x \ge 0 \end{matrix}\right.$ ,

and

$(c h)(x) = \left\{\begin{matrix} 0,& x< 0\\ (c p)x,& x \ge 0 \end{matrix}\right.$

$c h \in S$ . The second condition is met.

, as defined, is a subspace.

Report an Error

Example Question #389 : Linear Algebra

$C(-\infty , \infty)$ , the set of all continuous real-valued functions defined on $(-\infty, \infty)$ , is a vector space under the usual rules of addition and scalar multiplication.

Let be the set of all functions of the form

$f(x) = \left\{\begin{matrix} 2nx+2n , & x \le - 1 \\ 0, & -1 < x < 1\\ -nx+n & x \ge 1 \end{matrix}\right.$

for some real

True or false: is a subspace of $C(-\infty , \infty)$ .

Possible Answers:

False

True

Correct answer:

True

Explanation:

A set is a subspace of a vector space if and only if two conditions hold, both of which are tested here.

The first condition is closure under addition - that is:

If $a , b \in S$ , then $a+ b \in S$

Let $f , g \in S$ as defined. Then for some $n, m \in \mathbb{R}$ ,

$f(x) = \left\{\begin{matrix} 2nx+2n , & x \le - 1 \\ 0, & -1 < x < 1\\ -nx+n & x \ge 1 \end{matrix}\right.$

and

$g(x) = \left\{\begin{matrix} 2mx+2m, & x \le - 1 \\ 0, & -1 < x < 1\\ -mx+m & x \ge 1 \end{matrix}\right.$

Then

$(f+g)(x) = \left\{\begin{matrix}( 2nx+2n)+ ( 2mx+2m) , & x \le - 1 \\ 0+0 , & -1 < x < 1\\ (-nx+n) + (-mx+m) & x \ge 1 \end{matrix}\right.$

$(f+g)(x) = \left\{\begin{matrix}( 2nx+2mx )+ ( 2n +2m) , & x \le - 1 \\ 0+0 , & -1 < x < 1\\ (-nx+(-mx)) + (n+m) & x \ge 1 \end{matrix}\right.$

$(f+g)(x) = \left\{\begin{matrix} 2(n + m)x+ 2 ( n +m) , & x \le - 1 \\ 0 , & -1 < x < 1\\ -(n +m)x + (n+m) & x \ge 1 \end{matrix}\right.$

$f +g \in S$ . The first condition is met.

The second condition is closure under scalar multiplication - that is:

If $a \in S$ and is a scalar, then $ca \in S$

Let $h \in S$ as defined. Then for some $p \in \mathbb{R}$ ,

$h(x) = \left\{\begin{matrix} 2px+2p , & x \le - 1 \\ 0, & -1 < x < 1\\ -px+p & x \ge 1 \end{matrix}\right.$

For any scalar ,

$c h(x) = \left\{\begin{matrix} c (2px+2p) , & x \le - 1 \\ c \cdot 0, & -1 < x < 1\\ c(-px+p) & x \ge 1 \end{matrix}\right.$

$(c h)(x) = \left\{\begin{matrix} 2(cp)x+2 (c p) , & x \le - 1 \\ 0, & -1 < x < 1\\ -(cp)x+cp & x \ge 1 \end{matrix}\right.$

$c h \in S$ . The second condition is met.

, as defined, is a subspace.

Report an Error

Example Question #390 : Linear Algebra

If is an $m \times n$ matrix, find

dim(row(A))+dim(col(A))+dim(null(A))+dim(null(A^T)).

Possible Answers:

|n-m|

n+m

2n+2m

Correct answer:

n+m

Explanation:

Since a basis for the row space and the column space of a matrix have the same, number of vectors then their dimensions are the same, say .

By the rank-nullity theorem, we have rank(A) +Null(A) = n , or same to say

dim(row(A))+dim(null(A)) = n .

r +dim(null(A)) =n .

Hence dim(null(A))=n-r .

Finally, applying the rank-nullity theorem to the transpose of , we have

rank(A^T) +Null(A^T) = m , or the same to say

dim(row(A^T)) +dim(null(A^T)) = m .

r + dim(null(A^T))= m (The row space dimension of is the same as its transpose.)

dim(null(A))=r-m .

Adding all four of our findings together gives us

r+r+n-r+m-r = n+m .

Report an Error

← Previous 1 2 Next →

View Linear Algebra Tutors

Elvis
Certified Tutor

Methodist University, Bachelor of Science, Mathematics and Computer Science.

View Linear Algebra Tutors

Garth
Certified Tutor

View Linear Algebra Tutors

Dhruvil
Certified Tutor

Indus University, Bachelor of Science, Computer and Information Sciences, General. Concordia University-Ann Arbor, Master of ...

All Linear Algebra Resources

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

Popular Subjects

Computer Science Tutors in Los Angeles, LSAT Tutors in Chicago, Spanish Tutors in Seattle, Physics Tutors in Seattle, SSAT Tutors in Chicago, English Tutors in Philadelphia, MCAT Tutors in Washington DC, French Tutors in Seattle, LSAT Tutors in Seattle, Physics Tutors in San Diego

Popular Courses & Classes

SSAT Courses & Classes in Denver, GMAT Courses & Classes in Dallas Fort Worth, SAT Courses & Classes in Houston, GRE Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Chicago, LSAT Courses & Classes in San Diego, ACT Courses & Classes in Los Angeles, ACT Courses & Classes in New York City, GMAT Courses & Classes in Miami, GRE Courses & Classes in Houston

Popular Test Prep

ACT Test Prep in New York City, MCAT Test Prep in Chicago, LSAT Test Prep in Philadelphia, GMAT Test Prep in New York City, LSAT Test Prep in Boston, LSAT Test Prep in Phoenix, SSAT Test Prep in Los Angeles, SSAT Test Prep in Washington DC, SAT Test Prep in Chicago, ISEE Test Prep in San Francisco-Bay Area

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 : Range and Null Space of a Matrix

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #11 : Range And Null Space Of A Matrix

Example Question #11 : Range And Null Space Of A Matrix

Example Question #383 : Linear Algebra

Example Question #11 : Range And Null Space Of A Matrix

Example Question #385 : Linear Algebra

Example Question #386 : Linear Algebra

Example Question #11 : Range And Null Space Of A Matrix

Example Question #388 : Linear Algebra

Example Question #389 : Linear Algebra

Example Question #390 : Linear Algebra

All Linear Algebra Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: