Activity 7: Determinants and Volume MATH 250, Fall 2010 October 22, 2010 Names: In Activity 3, we explored the effect of a linear transformation T : R 2 fi R 2 on the Loras "L", where the standard matrix for T was A = a b c d . Let's revisit one of the problems from that assignment, reworded to fit the present worksheet: 1. a) What happens geometrically when you multiply an arbitrary vector by 1 0 0 2 ? (The diagram is below if you need it.) b) If A = 1 0 0 2 , what is T HX1, 0\)? What is T HX0, 1\)? (Remember, T HX1, 0\) is just alternate notation for T 1 0 .) c) The rectangle determined by the vectors T HX1, 0\) and T HX0, 1\) is the rectangle with vertices H0, 0L, T H1, 0), T H0, 1), and T H1, 0L + T H0, 1L. What is the area of this rectangle?
Transcript
Activity 7: Determinants and Volume
MATH 250, Fall 2010October 22, 2010
Names:
In Activity 3, we explored the effect of a linear transformation T : R2 ® R
2 on the Loras "L", where
the standard matrix for T was A =a b
c d. Let's revisit one of the problems from that assignment,
reworded to fit the present worksheet:
1. a) What happens geometrically when you multiply an arbitrary vector by 1 0
0 2? (The
diagram is below if you need it.)
b) If A =1 0
0 2, what is THX1, 0\)? What is THX0, 1\)? (Remember, THX1, 0\) is just
alternate notation for T1
0.)
c) The rectangle determined by the vectors THX1, 0\) and THX0, 1\) is the rectangle with vertices H0, 0L, TH1, 0), TH0, 1), and TH1, 0L + TH0, 1L. What is the area of this rectangle?
a 1
b 0
c 0
d 1.
K 1 00 1.
O
2 Determinants and volume.nb
2. a) Consider a transformation T with the standard matrix A =3 0
0 5. What are THX1, 0\L
and THX0, 1\L?
b) How does the area of the square determined by X1, 0\ and X0, 1\ change under the transformation T? That is, what is the area of the rectangle determined by the vectors THX1, 0\L and THX0, 1\L?
c) How does this relate to det A?
d) What if A =3 0
0 -5? How does the area of the rectangle determined by
THX1, 0\L =3
0 and THX0, 1\L =
0
-5 compare to det A?
3. a) More generally, suppose A =a 0
0 d. Write a formula for the area of the rectangle
determined by THX1, 0\L and THX0, 1\L under the transformation THxL = A x.
b) Let L be the set of points defining the Loras "L". That is, L is the set of yellow and purple points in the lefthand diagram above. It can also be thought of as the set of vectors representing those points. Using the standard matrix from part a), how do you think the area of THLL compares to the area of L? (THLL is the set of all images of points in L under the transformation THxL = A x.)
4. a) Now let A =a b
0 d. The effect of this transformation is just a horizontal shear of the
transformation determined by a 0
0 d. You can see an example of this by setting a = 1, d = 2, and
dragging the slider for b back and forth in the diagram above. How do the area of the rectangle
determined by a
0 and
0
d and the area of the parallelogram determined by
a
0 and
b
d
compare?
b) Find a formula for the area of the parallelogram determined by a
0 and
b
d in terms of
the determinant of A.
5. a) We saw in class yesterday that any square matrix can be reduced to an upper triangular matrix by row replacements and row swaps (we only need row scaling if we want reduced echelon form). What effect do row replacements and swaps have on the determinant of a matrix?
b) Finally, suppose A =a b
c d. State the general formula for the area of the parallelogram
determined by T HX1, 0\L =ac
and T HX0, 1\L =b
d in terms of det A.
c) Using the standard matrix from part b), how do you think the area of THLL compares to the area of L?
d) Find the formula for the area of a triangle with vertices at H0, 0L, ac
, and b
d.
Determinants and volume.nb 3
4. a) Now let A =a b
0 d. The effect of this transformation is just a horizontal shear of the
transformation determined by a 0
0 d. You can see an example of this by setting a = 1, d = 2, and
dragging the slider for b back and forth in the diagram above. How do the area of the rectangle
determined by a
0 and
0
d and the area of the parallelogram determined by
a
0 and
b
d
compare?
b) Find a formula for the area of the parallelogram determined by a
0 and
b
d in terms of
the determinant of A.
5. a) We saw in class yesterday that any square matrix can be reduced to an upper triangular matrix by row replacements and row swaps (we only need row scaling if we want reduced echelon form). What effect do row replacements and swaps have on the determinant of a matrix?
b) Finally, suppose A =a b
c d. State the general formula for the area of the parallelogram
determined by T HX1, 0\L =ac
and T HX0, 1\L =b
d in terms of det A.
c) Using the standard matrix from part b), how do you think the area of THLL compares to the area of L?
d) Find the formula for the area of a triangle with vertices at H0, 0L, ac
, and b
d.
6. a) Moving to three dimensions, how does the volume of the cube determined by the standard basis vectors X1, 0, 0\, X0, 1, 0\, and X0, 0, 1\ change under the transformation given by
a 0 0
0 b 0
0 0 c
? That is, what is the volume of the rectangular prism determined by Xa, 0, 0\, X0, b, 0\,
and X0, 0, c\?
b) By going through a similar process to that of problems 3-5, we can prove the following statement. If A is a 3 ´ 3 matrix, the volume of the parallelepiped determined by the columns of A is equal to the absolute value of the determinant of A. Using this formula, compute the volume of
the parallelepiped determined by the vectors
1
-2
0
,
-2
4
5
, and
3
-2
-1
.
• Let S be any region in R2 with finite area, and let T : R2 ® R
2 be the linear transformation determined by a 2 ´ 2 matrix A. By approximating the planar region S with a union of squares, we can show the following formula:
8area of THSL< = det A 8area of S<
7. Check to see that your answers to 3. b) and 5. c) match this formula.
• Let S be any region in R3 with finite volume, and let T : R3 ® R
3 be the linear transformation determined by a 3 ´ 3 matrix A. By approximating the planar region S with a union of cubes, we can show the following formula:
8volume of THSL< = det A 8volume of S<
4 Determinants and volume.nb
• Let S be any region in R2 with finite area, and let T : R2 ® R
2 be the linear transformation determined by a 2 ´ 2 matrix A. By approximating the planar region S with a union of squares, we can show the following formula:
8area of THSL< = det A 8area of S<
7. Check to see that your answers to 3. b) and 5. c) match this formula.
• Let S be any region in R3 with finite volume, and let T : R3 ® R
3 be the linear transformation determined by a 3 ´ 3 matrix A. By approximating the planar region S with a union of cubes, we can show the following formula:
8volume of THSL< = det A 8volume of S<
8. Consider the region E in R2 bounded by the ellipse with the equation x2
a2+
y2
b2= 1. That is, E
is the set of all points satisfying x2
a2+
y2
b2£ 1. We want to derive a formula for the area of the ellipse.
Let D be the unit disk. We claim that E = THDL, the image of the unit disk under the linear
transformation T with standard matrix A =a 0
0 b. You can explore the effects of the
transformation in the diagram below.
a) If a 0
0 b
u1
u2=
xy
, what is u1? What is u2?
b) Let Hu1, u2L be any point in the unit disk. That is, u12 + u2
2 £ 1. What inequality do you get when you substitute the values you found in part a) for u1and u2? What does this tells us about the point Hx, yL?
Similarly, if Hx, yL satisfies I xa
M2+ I y
bM2
£ 1, then u12 + u2
2 £ 1. Thus, the point Hu1, u2L is in D
if and only if its image is in E, proving the claim.
c) Therefore, the area of the ellipse is equal to the area of THDL. Using the formula above problem 7, find the area of THDL.
Determinants and volume.nb 5
a 1
b 1
K 1 00 1
O
6 Determinants and volume.nb
9. Let T : R3 ® R
3 be the linear transformation determined by the matrix A =
a 0 0
0 b 0
0 0 c
, where
a, b, and c are positive numbers. Let S be the unit ball, whose bounding surface has the equation x2 + y2 + z2 = 1. The sets S and THSL are represented in the diagram below.
a) Show that THSL is bounded by the ellipsoid with the equation x2
a2+
y2
b2+
z2
c2= 1.
b) Use the fact that the volume of the unit ball is 4 Π � 3 and the formula below problem 7 to determine the volume of THSL.
