/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q45E Consider an n 脳 m matrix A with... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Q45E (page 121)

Consider an n 脳 m matrix A with rank(A) = r < m. Explain how you can write ker(A) as the span of m 鈭� r vectors.

Short Answer

Expert verified

Short Answer Missed in the document

Step by step solution

01

Step 1:The kernel of a linear transformation

ThekernelofalineartransformationTx鈫�=Ax鈫�from鈩�mto鈩�nisthesoluctionsetofthelinersystemAx鈫�=0.

02

Proving ker(A) as the span of m − r vectors.

Consider an nxm matrix A with; rank (A)=r<m. By the definition of the kernel of a linear transformation, ker(A) is the solution set of the linear system

Since A is an nxm matrix with rank (A)=r<m.

Therefore, the solution set of A must have non-leading or free variables. Since, the kernel consists of zero-valued variables (vectors). Therefore, any vector in ker (A) can be expressed as a linear combination of them-n non-leading or free variables.

Hence, the definition of spanker(A) can be written as a span of m-r vectors.

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 91影视!

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

Consider the vectorsu鈫�,v鈫�andw鈫�sketched in the accompanying figure. Find the coordinate vector of w鈫�with respect to the basisu鈫�,v鈫�.

Two subspacesV andW of鈩�n are called complements if any vectorx鈫� in鈩�n can be expressed uniquely as x鈫�=v鈫�+w鈫�, wherev鈫� in V andw鈫� is in W. Show thatV andW are complements if (only if) canV鈭�W={0鈫�} and dim(V)+dim(W)=n.

Question: Consider linearly independent vectors v1鈫�,v2鈫�,....,vm鈫�in 鈩�nand let A be an invertible m脳mmatrix. Are the columns of the following matrix linearly independent?

[IV1IV2...IVmIII]A

We are told that a certain5脳5 matrixA can be written as

A=BC,

where Bis 5脳4andC is 4脳5. Explain how you know thatA is not invertible.

In Exercises 25through 30, find the matrix B of the linear transformationT(x鈫�)=Ax鈫� with respect to the basis 貜=(V1鈫�,--Vm鈫�) .

A=(1234);v鈫�1=[11,V鈫�2=12]
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.