MA-305 Bonus Homework 7

Due at noon, Friday, December 5, 1997
Extended to 1:00pm, Monday, December 8, 1997

Solution

You may do the calculations necessary for these problems either by hand or with Maple. Please submit a handwritten solution, or email an ASCII/Postscript/html document to the TA. (You may also submit through email which attached your Maple worksheet.)

1. Use QR factorization to find the least square fit of problem 4 in Homework 6.

   Solution:

   C(n) = 1.387891671 x ln( x ) - 0.955521423 x + 0.56617708.
   Maple worksheet(Maple text)

2. *Which of the following are linear transformations?
   (a) L( x, y, z) = ( 0, 0).
   (b) L( x, y, z) = (1, 2, -1).
   (c) L( x, y, z) = (x² + y, y - z).

   Solution:

   (a) Yes.
   (b) No.
   

   L( x₁, y₁, z₁) + L( x₂, y₂, z₂) = (1, 2, -1) + (1, 2, -1) = (2, 4, -2)
   not equivalent to L( x₁ + x₂, y₁ + y₂, z₁ + z₂) = (1, 2, -1).
   
   (c) No.
   
   L(1, 1, 1) + L(1, 1, 1) = (1² + 1, 1 - 1) + (1² + 1, 1 - 1) = ( 2, 0)
   not equivalent to L(1 + 1, 1 + 1, 1 + 1)= L(2, 2, 2) = (2² + 2, 2 - 2) = (6, 0).

3. **For the given matrix:

   [	5	0	-4	]
   [	0	-3	0	]
   [	-4	0	-1	]

   
   

   (a) Find its characteristic polynomial (in factor form).
   (b) Find its eigenvalues.
   (c) Find a basis for its eigenspace.

   Solution:

   (a) - ( lambda - 7) ( lambda + 3 )².
   (b) 7, -3.
   (c) {(-2, 0, 1)} when lambda = 7 ; {(1, 0, 2), (0, 1, 0)} when lambda =-3.

4. Which of the following are inner products? For those that are, decide the norm defined in the corresponding inner product.
   (a) < , >_a : R³ × R³ -> R
   

   	[	x1	]		[	x2	]
   <	[	y1	]	,	[	y2	]	>a ->	x1x2 + 2 y1y2 + 3 z1z2
   	[	z1	]		[	z2	]

   (b) < , >_b : R³ × R³ -> R
   

   	[	x1	]		[	x2	]
   <	[	y1	]	,	[	y2	]	>b ->	x1z2 + y1y2 + z1x2
   	[	z1	]		[	z2	]

   Solution:

   (a) Yes. || (x, y, z) || = ( x² + 2y² + 3z² )^1/2.
   (b) No. When v^T = [ 1, 2^1/2, -1 ] != 0, < v, v >_b = 0.

5. 	[	1	2	1	0	]
   A =	[	2	3	4	7	]	. Let LA(v) = A·v. Find:
   	[	3	5	5	7	]
   	[	3	4	7	14	]

   
   
   (a) the kernel of L_A. ( All v such that L_A(v) = 0. )
   (b) the range of L_A.( All w = L_A(v) for some v. )

   Solution:

   (a) The null space of A, the vector space spanned by {(-5, 2, 1, 0), (-14, 7, 0, 1)}.
   (b) The column space of A, Span([1, 2, 3, 3]^T, [2, 3, 5, 4]^T, [1, 4, 5, 7]^T, [0, 7, 7, 14]^T).

6. (Not covered)**Determine if the given matrix is diagonalizable. If so, diagonalize the matrix:

   [	3	3	]
   [	4	2	]

7. **(a) Find the general solution to the system of differential equations.
   (b) Find the particular solution satisfies the given initial conditions.
   du / dt = u + 3v
   dv / dt = 4u - 3v
   u = 5, v = 1 when t = 0

   Solution:

   (a) The general solution:

   	[	1	]		[	3/2	]
   C1	[	-2	]	e-5t + C2	[	1	]	e3t

   (b) The particular solution satisfies the given initial condition:
   u = (7/8)*e^-5t + (33/8)*e^3t
   v = (11/4)*e^3t - (7/4)*e^-5t