9.3. EXERCISES 199
(b) T replaces the ith component of x with b times the jth component added to theith component.
(c) T switches two components.
Show these functions are linear and describe their matrices.
24. In Problem 23, sketch the effects of the linear transformations on the unit square inR2. Give a geometric description of an arbitrary invertible matrix in terms of productsof matrices of these special matrices in Problem 23.
25. Let u = (a,b) be a unit vector in R2. Find the matrix which reflects all vectors acrossthis vector.
u
Hint: You might want to notice that (a,b) = (cosθ ,sinθ) for some θ . First rotatethrough −θ . Next reflect through the x axis which is easy. Finally rotate through θ .
26. Let u be a unit vector. Show the linear transformation of the matrix I− 2uuT pre-serves all distances and satisfies(
I−2uuT )T (I−2uuT )= I.
This matrix is called a Householder reflection. More generally, any matrix Q whichsatisfies QT Q = QQT is called an orthogonal matrix. Show the linear transformationdetermined by an orthogonal matrix always preserves the length of a vector in Rn.Hint: First either recall, depending on whether you have done Problem 51 on Page106, or show that for any matrix A,
⟨Ax,y⟩=⟨x,AT y
⟩27. Suppose |x|= |y| for x,y ∈ Rn. The problem is to find an orthogonal transformation
Q, (see Problem 26) which has the property that Qx = y and Qy = x. Show
Q≡ I−2x−y|x−y|2
(x−y)T
does what is desired.
28. Let a be a fixed vector. The function Ta defined by Tav = a+v has the effect oftranslating all vectors by adding a. Show this is not a linear transformation. Explainwhy it is not possible to realize Ta in R3 by multiplying by a 3×3 matrix.
29. In spite of Problem 28 we can represent both translations and linear transformationsby matrix multiplication at the expense of using higher dimensions. This is done bythe homogeneous coordinates. I will illustrate in R3 where most interest in this isfound. For each vector v = (v1,v2,v3)
T , consider the vector in R4 (v1,v2,v3,1)T .
What happens when you do1 0 0 a1
0 1 0 a2
0 0 1 a3
0 0 0 1
v1
v2
v3
1
?