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Chapter 3: Q35E (page 120)

Consider a nonzero vector 蠀鈫in 鈩3.Arguing geometrically, describe the image and the kernel of the linear transformation Tfrom 鈩3to 鈩to given by,

role="math" localid="1659526111480" T(x鈫)=蠀鈫路x鈫.

Short Answer

Expert verified

The kernel of transformation consists of all x鈫鈭鈩3that are orthogonal to vectors蠀鈫and the image is the whole鈩.

Step by step solution

01

Step by Step Solution:  Step 1: To define kernel of linear transformation

The kernel of linear transformation is defined as follows:

The kernel of a linear transformation Tx鈫=Ax鈫from 鈩mto 鈩nconsists of all zeros of the transformation, i.e., the solutions of the equations Tx鈫=Ax鈫=0.

It is denoted by kerT or kerA.

We have given the linear transformation T:鈩3鈫鈩defined by Tx鈫=蠀鈫路x鈫.

02

To describe the image of the linear transformation

The image of Tconsists of all vectors represented as:

蠀1鈥鈥鈥鈥蠀2鈥鈥鈥蠀3in鈩3 , which are the column vectors.

Here, 蠀i鈭鈩which are dependent and any of them spans the whole鈩 .

03

To describe the kernel of the linear transformation

The kernel of transformation consists of all x鈫鈭鈩3that are orthogonal to vectors 蠀鈫.

Therefore, their linear combination is 0.

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Most popular questions from this chapter

Determine whether the following vectors form a basis of 鈩4; [1111],[1-11-1],[1248],[1-24-8].

In Exercises 1 through 20, find the redundant column vectors of the given matrix A 鈥渂y inspection.鈥 Then find a basis of the image of A and a basis of the kernel of A.

19. 10530014200000100000

In Exercises 21 through 25, find the reduced row-echelon form of the given matrix A. Then find a basis of the image of A and a basis of the kernel of A.

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Explain why you need at least 鈥榤鈥 vectors to span a space of dimension 鈥榤鈥. See Theorem 3.3.4b.

Consider a 5x4matrixA=[||||V鈫1V鈫2V鈫3V鈫4||||] . We are told that the vector[1234] is in the kernel of A. Writev鈫4 as a linear combination of V鈫1,V鈫2,V鈫3.
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