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Chapter 4: Q43E (page 177)

If matrix A represents the reflection about a line LinR2,what is the dimension of the space V of all matrices S such thatAS=S(100-1)?

Hint: WriteS=[v鈫�w鈫�] , and show that v鈫�must be parallel to L, while w鈫� must be perpendicular to L.

Short Answer

Expert verified

The dimensions of the space V of alln脳nmatrices S is 2.

Step by step solution

01

Explanation for the dimension of the space

If a linear space has a basis with n elements then all other bases of V, consists of n elements as well.

Also we say that n is the dimension of V

dim(V)=n

02

Step 2:Solution for the dimension of the space V

Consider, the matrix A which represents the reflection about a line L inR2

HereAS=S(100-1)is the matrix of the reflection about a line L

Here A is the2脳2matrix of the reflection about the line L.

Thusn=2in the matrices S of the reflection about the line L.

dim(V)=ndim(V)=2

Since the solution.

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