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Chapter 2: Q40E (page 73)

Let Pand Qbe the two perpendicular linesin鈩�2.For a vectorx鈫� in鈩�2 , whatisprojP(x鈫�)+projQ(x鈫�)?Give your answer in termsofx鈫�.Draw a sketch to justify your answer.

Short Answer

Expert verified

projP(x鈫�)+projQ(x鈫�)=x鈫�. The sketch representing the same is,

Step by step solution

01

Compute the linear transformation 

The relation between the linear transformation and the projection is,

T(x鈫�)=projL(x鈫�)

The above equation represents the linear transformation of the vectorx鈫� onto the line L.

02

Define the relation

The projection of the vector, that is,projP(x鈫�)represents the projection of the vector (x鈫�)onto the linePand projQ(x鈫�)represents the projection of the vector (x鈫�)onto the line .

Thus, the sum of these projections equal to the vector itself.

projP(x鈫�)+projQ(x鈫�)=x鈫�

HenceprojP(x鈫�)+projQ(x鈫�)=x鈫�,.

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Most popular questions from this chapter

TRUE OR FALSE?

There exists an invertible2脳2 matrix A such thatA-1=-A.

TRUE OR FALSE?

There exists an upper triangular2脳2 matrixAsuch that A2=(1101) .

Question: TRUE OR FALSE?

If T is any linear transformation from鈩�3to鈩�3 , thenT(v鈫�脳w鈫�)=T(v鈫�)脳T(w鈫�)for all vectors v鈫� andw鈫�in鈩�3.

If matrices A and B commute, then the formula A2B = BA2 must hold.

Use part (c) of Theorem 2.3.11 to prove part (b): If Ais a regular transition matrix of size n脳nwith equilibrium distribution x鈫�equ, and ifx鈫� is any distribution vector in鈩�n , then limm鈫�鈭�(Amx鈫�)=x鈫�equ.
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