Mathematical Tools for Physics

7—Operators and Matrices 201

Problems

7.1 Draw a picture of the effect of these linear transformations on the unit square with vertices at(0,0),(1,0),
(1,1),(0,1). The matrices representing the operators are

(a)

(

1 2

3 4

)

, (b)

(

1 − 2

2 − 4

)

, (c)

(

−1 2

1 2

)

Is the orientation preserved or not in each case? See the figure at the end of section7.6

7.2 Using the same matrices as the preceding question, what is the picture resulting from doing (a) followed by
(c)? What is the picture resulting from doing (c) followed by (a)? The results of section7.4may prove helpful.

(a,c)

(b,d)

(a+b,c+d)

7.3Look again at the parallelogram that is the image of the unit square in the calculation
of the determinant. In Eq. ( 26 ) I used the cross product to get its area, but sometimes a

brute-force method is more persuasive. If the transformation has components

(

a b

c d

)

The corners of the parallelogram that is the image of the unit square are at(0,0),(a,c),
(a+b,c+d),(b,d). You can compute its area as sums and differences of rectangles
and triangles. Do so; it should give the same result as the method that I used.

7.4 In three dimensions, there is an analogy to the geometric interpretation of the cross product as the area of a
parallelogram. The triple scalar productA~.B~×C~is the volume of the parallelepiped having these three vectors
as edges. Prove both of these statements starting from the geometric definitions of the two products. That is,
from theABcosθandABsinθdefinitions of the dot product and the magnitude of the cross product (and its
direction).

7.5 Derive the relation~v=~ω×~rfor a point mass rotating about an axis. Refer to the figure before Eq. ( 2 ).