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Math 2600 Homework Assignment Handouts Spring 2017
Collected Homework Number 1. Due Thursday January 19, 2017.
Read Section 1.113 and Appendix C, carely before attacking the problems
Section 1.1: Problems
HW 1 Solutions
Problem 3
Let A(n) denote the statement 1 + 3 + 5 + + (2n 1) = n2 .
We wish to show that A(n) is true for all positive integers n.
For n = 1, 1 = (1)2 so A(1) is true.
Suppose A(k ) is true. We want to show A(k + 1) is true. We calculate,
1
MATH 204
Quiz 2 Selected Solution
written by Clark Chong (fan.f.chong@vanderbilt.edu)
v1 = (1, 2, 3), v2 = (3, 2, 1). The given linear transformation
be A. Then the given conditions translate to Av1 = (2, 5, 6, 1)
and Av2 = (2, 3, 2, 1).
Note that (4, 4,
Quiz 1. August 31, 2011
1. The reduced row echelon form of the
1
0
0
augmented matrix of a system of linear equations is:
2 0 3 0 24
0 1 5 0 76
0 0 0 1 8
Find the general solution of the system of equations.
2. Complete the sentence:
A matrix has rank 1
MATH 204
Quiz 9 Solutions and Comments
prepared for you by Clark Chong (fan.f.chong@vanderbilt.edu)
2 There are at least three ways to solve this problem. Even if you got full score on this problem,
I would still strongly recommend you to check out the ot
MATH 204
Quiz 8 Solutions and Comments
prepared for you by Clark Chong (fan.f.chong@vanderbilt.edu)
1. Let v1 , , vn be the column vectors of A, i.e. A may be represented as [v1 vn ]. Then we
have V = Im(A) = Spancfw_v1 , , vn and hence
Im(A) = cfw_u : u
MATH 204
Quiz 7 Solutions and Comments
prepared for you by Clark Chong (fan.f.chong@vanderbilt.edu)
1. (The solution is divided into two parts, we will rst apply Gram-Schmidt to obtain an
orthogonal basis. Then we will normalize the basis. This way we can
MATH 204
Quiz 6 Solutions and Comments
prepared for you by Clark Chong (fan.f.chong@vanderbilt.edu)
1. (a) There are at least three ways to prove that B = cfw_1 + x, 2x is indeed a basis.
The most direct way would be to use the denition of a basis and che
MATH 204
HW 8 Solution and Comments to Selected Problems
prepared specically for you by Clark Chong (fan.f.chong@vanderbilt.edu)
5.3.29 Because orthogonal transformation preserves length of vector, hence we have the following:
(T v, T w) = cos1
Tv Tw
Tv T
MATH 204
HW 5 Selected Solution
prepared specically for you by Clark Chong (fan.f.chong@vanderbilt.edu)
34 (a) dim(V ) = # of free variables in A (the coecient matrix associated to the system
of homogeneous linear equations)
= # of variables # of leading
MATH 204
HW 4 Selected Solution
written by Clark Chong (fan.f.chong@vanderbilt.edu)
27 One such example is f (x) = x(x + 1)(x 1). Now we need to show it is surjective but
not invertible.
f is continuous, limx f (x) = and limx f (x) = hence im(f )=R.
f (x)