Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Prove theorem 3 as follows: Given an m×n matrix A, an element in ColA has the form Ax for some x in Rn. Let Ax and Aw represent any two vectors in ColA.

  1. Explain why the zero vector is in ColA.
  2. Show that the vector Ax+Aw is in ColA.
  3. Given a scalar c, show that c(Ax) is in ColA.

Short Answer

Expert verified

a. The zero vector is in ColA.

b. It is proved that the vector Ax+Aw is in ColA.

c. It is proved that c(Ax) is in ColA

Step by step solution

01

Explain that the zero vector is in ColA

a)

It is given that in an m×n matrix, an element in ColA has the form Ax for some x in Rn.

Thezero vector is in ColA because of the equation A0=0.

Thus, the zero vector is in ColA.

02

Show that the vector \(A{\mathop{\rm x}\nolimits}  + A{\mathop{\rm

b)

It is easy to find vectors in ColA. Thecolumnsof A are displayed; others are formed from them.

The vector Ax+Aw is in ColA becauseAx+Aw=A(x+w).

Thus, it is proved that the vector Ax+Aw is in ColA.

03

Show that c(Ax) is in ColA

c)

The vector c(Ax) is in ColA because c(Ax)=A(cx).

Thus, it is proved that c(Ax) is in ColA.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Study anywhere. Anytime. Across all devices.

Sign-up for free