Math 4133 - Linear Algebra
Exam 1 - 2016.03.31
YOUR NAME HERE
(remember Alt+7 gives a text bracket)

1. Find the area of the triangle whose vertices are located at the three points:
 P(-2, 3, 2, 5, 1), Q(1, 1, 0, 1, 3), and R(5, 2, -1, 1, 1).

P = {-2,3,2,5,1};
Q = {1,1,0,1,3};
R = {5,2,-1,1,1};

2. Compute the cross product of the two vectors: (OverVector[u]) = <-1, 2, 3>, and (OverVector[v]) = <2,0,1>.

u = {-1,2,3};
v = {2,0,1};

3. Consider the following four vectors:
(OverVector[(Subscript[u, 1])]) = <2, 3, 4, -1, 2>, (OverVector[(Subscript[u, 2])]) = <1, 0, 1, 1, 0>, (OverVector[(Subscript[u, 3])]) = <3, 2, -1, 4, 5>, and (OverVector[(Subscript[u, 4])]) =  <-6, -3, 8, -12, -13>.

(Subscript[u, 1]) = {2,3,4,-1,2};
(Subscript[u, 2]) = {1, 0, 1, 1, 0};
(Subscript[u, 3]) = {3,2,-1, 4, 5};
(Subscript[u, 4]) = {-6, -3, 8, -12, -13};

(a) Compute the dimension of the subspace spanned by the set {(OverVector[(Subscript[u, 1])]), (OverVector[(Subscript[u, 2])]), (OverVector[(Subscript[u, 3])]), (OverVector[(Subscript[u, 4])])}.

(b) Find a basis for the subspace spanned by the set {(OverVector[(Subscript[u, 1])]), (OverVector[(Subscript[u, 2])]), (OverVector[(Subscript[u, 3])]), (OverVector[(Subscript[u, 4])])}.

(c) Find a basis for the orthogonal subspace to the span of {(OverVector[(Subscript[u, 1])]), (OverVector[(Subscript[u, 2])]), (OverVector[(Subscript[u, 3])]), (OverVector[(Subscript[u, 4])])}.

(d) Compute the dimension of the subspace from part (c).

4. Consider the set of all continuous functions on the closed interval [-1,1], which is denoted by the symbol C[-1,1], and the set B = {1, x, (x^2), (x^3), (x^4), (x^5)} and the subspace spanned by the set B.  Furthermore, let the dot product of two functions (vectors) in C[-1,1] be defined by the following integral (similar to HW problems from Chapter 8):

                                              f(x)* g(x) = (∫_(-1)^1)f(x)g(x)dx
                                              
 (a) Perform the Gram-Schmidt Orthonormalization Process on the basis B, and verify this result by computing the dot product matrix W.
 (b) Project the function G(x) = (Tanh^2)(4x-1).
 (c) Graph both G(x) and the projection computed in part (b).

(Subscript[f, 1])= 1; 
(Subscript[f, 2])= x;
(Subscript[f, 3]) = x^2; 
(Subscript[f, 4]) = x^3;
(Subscript[f, 5]) = x^4;
(Subscript[f, 6])= x^5;

(b) Project the function G(x) = (Tanh^2)(4x-1)

(c) Graph both G(x) and the projection computed in part (b).