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Math 313/515, Spring 2005 Jerry L. Kazdan
Homework Set 7, Due Thursday, March 17, 2005 ( Late papers will be accepted until 4 PM on Fri. March 18 )
- a) Compute (by hand) the determinant of the following matrix:
A :=
by reducing it to the determinant of a lower triangular matrix. b) Compute the volume of the 4-dimensional “parallelogram” Q spanned by the vec- tors V 1 = ( 1 , − 1 , 0 , 2 ), V 2 = ( 2 , − 2 , − 1 , 5 ), V 3 = (− 1 , 3 , 4 , 0 ), V 4 = ( 0 , 1 , − 3 , 1 ). c) The matrix
B :=
can be thought of as a map of R^4 to itself. Compute (by hand) the volume of B ( Q ) (this is the image of Q under the map B ). Of course you can use any formulas, just not a computer or a fancy hand calculator.
- Compute (by hand) the determinant of the following matrix.
C :=
a 1 0 0 0 b 1 0 0 0 c 0 0 1 − b c 0 0 1 − a d e 1 f g
- Use Cramer’s rule to solve the equation AX = Y , where A is given below and Y is the column vector Y = ( y 1 , y 2 , y 3 ). Then observe you have computed A −^1 , so exhibit it.
A :=
- a) For what value(s) of the constant α are the vectors
X 1 = ( 1 , − 2 , 1 ), X 2 = ( 2 , α, 2 ), X 3 = ( 1 , 2 , 3 )
linearly dependent? b) For what value(s) of β does the system of equations
x + 2 y + z = 0 − 2 x + β y + 2 z = 0 x + 2 y + 3 z = 0
have more than the trivial solution x = y = z = 0? (Explain your answer.)
- Let ∆ n = det Mn be the determinant of an n × n matrix Mn with a ’s along the main diagonal and b ’s on the two “off diagonals” directly above and below the main diagonal (this is a simple example of a tridiagonal matrix). Thus
M 5 =
a b 0 0 0 b a b 0 0 0 b a b 0 0 0 b a b 0 0 0 b a
a) Prove ∆ n = a ∆ n − 1 − b^2 ∆ n − 2. b) Compute ∆ 1 and ∆ 2 by hand. Then use the formula to compute ∆ 3 and ∆ 4. REMARK: In a few weeks we will show how one can use part a). above to get a general formula for ∆ n ].
- After you write the following Matlab functions, please print a copy as part of the home- work. IN ADDITION, please send a copy as an email attachment sent to our TA: [email protected] so she can test it with some real input. a) Given an angle theta (in degrees), write a Matlab function R = rotz(theta) that returns a 3 × 3 matrix corresponding to a rotation in R^3 by theta degrees around the z axis. b) Write a Matlab function g = Transform(A,V) that takes a 3x3 matrix A and a vector V viewed as a 3 × 1 matrix and returns a 4 × 4 matrix, g = [A V; 0 0 0 1]. [This is the homogeneous version of the transformation generated by A and V as in Homework Set 2 #11].
- Let n ∈ R^3 be a unit vector. Find a formula for the 3 × 3 matrix that determines a rotation of R^3 through an angle θ with n as axis of rotation (assuming the axis passes through the origin). Here we outline how to find this formula — but before reading further, try finding it on your own. a) (Example) Find a matrix that rotates R^3 through the angle θ using the vector ( 1 , 0 , 0 ) as the axis of rotation. b) More generally, let u and w be orthonormal vectors in the plane perpendicular to n. Show that the map
R n : x 7 → ( x · n ) n + cos θ u + sin θ w
rotates x through an angle θ with n as axis of rotation. [Note: one needs more information to be able to distinguish between θ and −θ]. c) Using problem 2 to write u and w , in terms of n and x , show that one can rewrite the above formula as
R nx = ( x · n ) n + cos θ [ x − ( x · n ) n ] + sin θ ( n × x ) = x + sin θ ( n × x ) + ( 1 − cos θ)[( x · n ) n − x ].
Thus, using problem 1, if n = ( a , b , c ) ∈ R^3 deduce that:
R n = I + sin θ
0 − c b c 0 − a − b a 0
(^) + ( 1 − cos θ)
− b^2 − c^2 ab ac ab − a^2 − c^2 bc ac bc − a^2 − b^2
d) Let A n be as in Problem 1 (but using n rather than v ). Show that
R n = I + sin θ A n + ( 1 − cos θ) A^2 n.
e) Use this formula to find the matrix that rotates R^3 through an angle of θ using as axis the line through the origin and the point ( 1 , 1 , 1 ).
