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Chapter 2: Q3E (page 107)

The function T[xy]=[x-yy-x]is a linear transformation?

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Definition of linear Transformation

Therefore, from Definition- A function T fromRm to Rnis called a linear transformation if there exists ann脳m matrix A such that T(x鈫�)=Ax鈫� for all x in the vector spaceRm .

02

Check for the given statement

Notice that the given function can be representing as follows:

Txy=1-1-11xy

Hence, T is a linear transformation.

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Most popular questions from this chapter

Question:

  1. If Ais any 3x3 transition matrix (see Definition 2.1.4), find the matrix product [111]A .
  2. For a fixed n , let e鈫� be the row vector e鈫�=[1,1,..,1]n1's . Show that an nxn matrix A with nonnegative entries is a transition matrixif (and only if)e鈫�A=e鈫� .

TRUE OR FALSE?

There exists a real number Ksuch that the matrix[k-1-2-4k-3]fails to be invertible.

TRUE OR FALSE?

If a matrixA=(abcd) represents the orthogonal projection onto a line L , then the equationa2+b2+c2+d2=1must hold.

Consider a linear transformation Tfrom R2to R2. Suppose thatv鈫�and w鈫� are two arbitrary vectors in R2and thatx鈫�is a third vector whose endpoint is on the line segment connecting the endpoints ofv鈫�and w鈫�. Is the endpoint of the vectorT(x)鈫�necessarily on the line segment connecting the endpoints ofT(v)鈫�and T(w)鈫�? Justify your answer.

Question 7: Suppose v1鈫�,v2鈫�,v3鈫�鈰�vm鈫�are arbitrary vectors in Consider the linear transformation from togiven by

T[x1x2鈰�xm]=[x1v1鈫�,x2v2鈫�,x3v3鈫�鈰�xmvm鈫�]

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