Call Now to Set Up Tutoring:

(888) 888-0446

Linear Algebra : Operations and Properties

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 #19 : Range And Null Space Of A Matrix

$C(-\infty , \infty)$ , 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:

True

False

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 #12 : Range And Null Space Of A Matrix

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

Example Question #311 : Operations And Properties

Calculate the determinant of matrix A where

$A = \begin{bmatrix} 5&4 \\ 1&2 \\ 0&7 \end{bmatrix}$

Possible Answers:

Not Possible

-50

Correct answer:

Not Possible

Explanation:

The matrix must be square to calculate its determinant, therefore, it is not possible to calculate the determinant for this matrix.

Report an Error

Example Question #312 : Operations And Properties

Calculate the determinant of matrix A where,

$A=\begin{bmatrix} 4&1 \\ 5& 3 \end{bmatrix}$

Possible Answers:

-7

Correct answer:

Explanation:

To calculate the determinant of a 2x2 matrix, we can use the equation $a_{^{11}}a_{^{22}}-a_{^{12}}a_{^{21}}$

Report an Error

Example Question #313 : Operations And Properties

Calculate the determinant of matrix A where,

$A=\begin{bmatrix} 8&-8 \\ 51&12 \end{bmatrix}$

Possible Answers:

-315

-504

504

Correct answer:

504

Explanation:

To calculate the determinant of a 2x2 matrix, we can use the equation $a_{^{11}}a_{^{22}}-a_{^{12}}a_{^{21}}$

Report an Error

Example Question #314 : Operations And Properties

Calculate the determinant of matrix A where,

$A=\begin{bmatrix} 8&6 \\ 0&2 \end{bmatrix}$

Possible Answers:

-15

Correct answer:

Explanation:

To calculate the determinant of a 2x2 matrix, we can use the equation $a_{^{11}}a_{^{22}}-a_{^{12}}a_{^{21}}$

Report an Error

Example Question #315 : Operations And Properties

Calculate the determinant of matrix A where,

$A=\begin{bmatrix} 2&5 &6 \\ 2&1 &0 \\ 5&3 &4 \end{bmatrix}$

Possible Answers:

-26

-24

Correct answer:

-26

Explanation:

Calculating the determinant of a 3x3 matrix is more difficult than a 2x2 matrix. To calculate the determinant of a 3x3 matrix, we use the following $\left | A\right |=a_{11}(a_{22}a_{33}-a_{23}a_{32})-a_{12}(a_{21}a_{33}-a_{23}a_{31})+a_{13}(a_{21}a_{32}-a_{22}a_{31})$

Report an Error

Example Question #316 : Operations And Properties

Calculate the determinant of matrix A where,

$A=\begin{bmatrix} -5&7 &10 \\ -20&15 &2 \\ 0& 3& 8 \end{bmatrix}$

Possible Answers:

-50

-49

Correct answer:

-50

Explanation:

Report an Error

Example Question #317 : Operations And Properties

Calculate the determinant of $\begin{bmatrix} 5&-2 \\ 1&3 \end{bmatrix}$ .

Possible Answers:

Correct answer:

Explanation:

By definition,

$\begin{vmatrix} a&b \\ c&d \end{vmatrix}= a(d)-b(c)$ ,

therefore,

$\begin{vmatrix} 5&-2 \\ 1&3 \end{vmatrix}= 5(3)-1(-2)= 17$ .

Report an Error

Example Question #318 : Operations And Properties

Calculate the determinant of $\begin{bmatrix} 1&3 &1 \\ 0& 2&0 \\ 4&8 &2 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

For simplicity, we will find the determinant by expanding along the second row. Consider the following:

$\begin{vmatrix} 1&3 &1 \\ 0& 2&0 \\ 4&8 &2 \end{vmatrix}= -0[3(2)-1(8)]+2[1(2)-1(4)]-0[1(8)-3(4)]= -4$

Report an Error

View Linear Algebra Tutors

Lakshay
Certified Tutor

Xavier University, Bachelor, Mechanical engineering .

View Linear Algebra Tutors

Megan
Certified Tutor

University of Wisconsin-Madison, Bachelor of Science, Chemical Engineering.

View Linear Algebra Tutors

David
Certified Tutor

The University of Texas at El Paso, Bachelor of Science, Computational Mathematics. Indiana State University, Master of Scien...

All Linear Algebra Resources

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

Popular Subjects

ISEE Tutors in Phoenix, ISEE Tutors in San Diego, English Tutors in Philadelphia, French Tutors in Dallas Fort Worth, Computer Science Tutors in Miami, SAT Tutors in San Diego, ISEE Tutors in Seattle, GMAT Tutors in Boston, Math Tutors in San Francisco-Bay Area, Statistics Tutors in Houston

Popular Courses & Classes

GMAT Courses & Classes in Seattle, ISEE Courses & Classes in Denver, ISEE Courses & Classes in Boston, Spanish Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in New York City, GRE Courses & Classes in San Diego, GMAT Courses & Classes in Denver, GMAT Courses & Classes in New York City, ACT Courses & Classes in Atlanta, GMAT Courses & Classes in Dallas Fort Worth

Popular Test Prep

GMAT Test Prep in Boston, GRE Test Prep in Dallas Fort Worth, SSAT Test Prep in Miami, ACT Test Prep in Phoenix, GMAT Test Prep in Philadelphia, SAT Test Prep in New York City, MCAT Test Prep in New York City, LSAT Test Prep in Chicago, GMAT Test Prep in Seattle, ISEE Test Prep in Houston

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 : Operations and Properties

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #19 : Range And Null Space Of A Matrix

Example Question #12 : Range And Null Space Of A Matrix

Example Question #311 : Operations And Properties

Example Question #312 : Operations And Properties

Example Question #313 : Operations And Properties

Example Question #314 : Operations And Properties

Example Question #315 : Operations And Properties

Example Question #316 : Operations And Properties

Example Question #317 : Operations And Properties

Example Question #318 : 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: