Math 4133 - Linear Algebra
Exam 3 - 2015.05.01
YOUR NAME HERE
(remember Alt+7 gives a text bracket)

1. A linear map T is defined by the following four points. Determine the matrix A that satisfies T((OverVector[v])) = A(OverVector[v]).

T(⟨1, -2, 1, -1⟩) = ⟨1, 2, 1, 0⟩
T(⟨1,0,1,1⟩) = ⟨1, 0, 1, 1⟩
T(⟨1,2,1,0⟩) = ⟨0,1,1,1⟩
T(⟨0,1,1,1⟩) = ⟨1,-2,1,-1⟩

D[Sin[x],x]

2. The following matrix A represents a linear map T:(R^3) ⟶ (R^8).
A = (({{1, -3, -5}, {0, 1, 3}, {-1, 7, 2}, {3, 3, -1}, {-1, 0, 5}, {-1, 8, 3}, {6, 3, -1}, {-1, -9, 0}}))
(a) Compute Ker(T), expressing your answer in terms of a basis.
(b) Compute Im(T), expressing your answer in terms of a basis.
(c) Compute the dimensions of both Ker(T) and Im(T).
(d) Determine if T is one-to-one or onto.  Explain your answers!

D[Cos[x],x]

3. Construct a map T:(R^4) ⟶ (R^4) which takes vectors in (R^4) expressed in terms of basis (Subscript[B, 1]) and expresses them in terms of the basis (Subscript[B, 2]) as given below:

(Subscript[B, 1]) : { (Subscript[(OverVector[v]), 1]) = ⟨1, 1, -1, 0⟩,  (Subscript[(OverVector[v]), 2]) = ⟨0, 1, -1, 1⟩,  (Subscript[(OverVector[v]), 3]) = ⟨1, 0, -1, -1⟩,  (Subscript[(OverVector[v]), 4]) = ⟨1, 0, 1, 0⟩ }
(Subscript[B, 2]) : { (Subscript[(OverVector[w]), 1]) = ⟨1, 0, 1, 1⟩,  (Subscript[(OverVector[w]), 2]) = ⟨0, 1, 1, 1⟩,  (Subscript[(OverVector[w]), 3]) = ⟨0, 0, 1, 0⟩,  (Subscript[(OverVector[w]), 4]) = ⟨-1, -1, 1, 0⟩ }

Note: You may express the map T in terms of matrix multiplication by a 4x4 matrix A.

D[Tan[x],x]

4. For the dataset (given in (x,y) pairs) S = {{-0.1,2.3}, {-0.4,5.3}, {0.2,1.5}, {0.4,-1.3}, {-0.1,2.2}, {-0.4,5.1}, {0.3,-1.7}, {0.7,-4.3}, {0.6,-3.3}, {-0.25,3.5}}, find the line y = m x + b of best fit the data.  Compute the error in your curve fit by computing |A(OverVector[y])-(OverVector[b])|.  Plot both the points and the line of best fit together.

D[Exp[x],x]

5. For the dataset  (given in (x, y, z) triples)  S = {{1935,14.4,18.7}, {2233,14,21.7}, {2235,14.1,24.2}, {1970,14.2,21.6}, {2254,16.3,19.3}, {2238,16.3,20.2}, {2337,15.6,20.6}, {1951,16,8.6}, {1419,15.4,2.8}, {1295,13.9,9.8}, {2592,14.2,21.7}, {2592,17.1,7.1}, {1885,15.4,6.7}, {1361,14.8,4}, {1037,17,2.7}, {1419,18,4.1}, {1345,16.4,3}, {1334,16.9,3.1}, {1110,16.6,0.7}, {668,11.5,8.5}, {730,11.6,4.7}, {785,11.9,14.2}}, find the surface response curve z = a(x^2) + b(y^2) + cxy + dx + ey + f of best fit the data.  Compute the error in your curve fit by computing |A(OverVector[z])-(OverVector[b])|. Plot both the points the the surface response curve of best fit together.

D[1/x,x]