Solutions to Class 20

  1. Suppose that f is the function from R3 to R3 which rotates every vector A counterclockwise about some axis L.

    a) Show that f is a linear transformation.

    Solution

    In order to show f is a linear transformation, it must be proven that:
    f(A+B)= f(A) + f(B)
    figure 1
    Rotating vectors A, B, and A+B counterclockwise through an arbitrary angle, we get two similar triangles, and it is obvious
    f(A+B)= f(A) + f(B)

    For a linear transformation, it must also be shown:

    f(kA)= k*f(A)
    figure 2
    This can be seen in the figure above. Thus, f is a linear transformation.

    b) Show that the column-vectors in the standard matrix A of f are pairwise orthogonal and their lengths are equal to 1.

    Solution

    The first column-vector in the standard matrix A is obtained by rotating the x axis (1,0,0) counterclockwise about L. The length will not change, and is equal to 1. The second column-vector is found by the same procedure as (0,1,0). The length will also be equal to 1.

    Looking at the x-y plane, the x-axis rotates to x', and the y-axis rotates to y'. x' and y' are still perpendicular to each other, and they still form a plane. Thus, the first and second column vectors are orthogonal. This procedure also holds for x and z, as well as y and z, so the column-vectors are pairwise orthogonal, and equal to one.

  2. Find the standard matrix of the linear transformation of R3 which reflects every vector V about the plane
    ax+by+cz=0

    Solution

    The normal vector of the plane, n, is (a, b, c).
    figure 1
    >From the figure we see the reflection of V, V':
    V'= -U+P
    where U is the projection vector on n. Let U=(ta, tb, tc), and V=(1, 0, 0). Thus, <U-V, U>= 0.
    ta(ta-1)+ t2b2+ t2c2=0
    
                               a
                       t = ---------
                           (a2+b2+c2)

                     a2       ab        ac 
             U = (-------, --------, --------)
                  a2+b2+c2  a2+b2+c2   a2+b2+c2

                        a2         ab          ac
       P = V-U = (1- -------, - --------, - --------)
                     a2+b2+c2    a2+b2+c2     a2+b2+c2

                       2a2        2ab         2ac
     V' = -U+P = (1- -------, - --------, - --------)
                     a2+b2+c2    a2+b2+c2     a2+b2+c2 

	                  1 
            V' = ------- (b2+c2-a2, -2ab, -2ac)
                 a2+b2+c2
    This is the first column of the standard matrix A. Now let V= (0,1,0). Similarly, 
                        1
            V' = ------- (-2ab, a2+c2-b2, -2bc)
                 a2+b2+c2
    is the second column of A. Finally, let V= (0,0,1). 
                        1
            V' = ------- (-2ac, -2bc, a2+b2-c2)
                 a2+b2+c2
    is the third column of A. So the standard Matrix A is: 
                        1    ( b2+c2-a2  -2ab    -2ac  )
            A  = ------- (   -2ab  a2+c2-b2  -2bc  )
                 a2+b2+c2 (   -2ac   -2bc   a2+b2-c2)

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