Exam Questions

1. Given (OverVector[v])= ⟨1-i, 2+3i, 6+5i, 2-3i⟩, and  (OverVector[w])= ⟨-1+2i, -4-6i, 6+2i, 4+4i⟩, compute the following:
(a) (OverVector[v])·(OverVector[w])
(b) (OverVector[w])·(OverVector[v])
(c) |(OverVector[w])|
(d) |(OverVector[v])|

2. Convert the following matrix to upper triangular form by left multiplying be elementary matrices.
A = (({{2, 1, -1, 3, 1}, {1, 4, 0, 0, 1}, {-2, 3, 6, 1, 2}, {1, 6, -4, -2, -8}, {3, -2, 7, 5, -1}}))

3. Diagonalize the matrix A = (({{1, 0, 0, 1}, {0, 1, 1, 0}, {1, 0, -1, 1}, {1, 1, 1, 0}})).  Then use this diagonalization to compute (e^(A t)).

4. Consider the following dataset:

{(-1.98,-2.13), (-1.45, -3.65), (-1.11, -4.24), (-1.03,0.23), (-0.05,0.92), (0.01,-4.67), (1.03,0.92),(0.97,-5.02), (2.1,0.87), (1.97,-4.78), (3.01,-4.24), (3.1,0.3), (3.5,-0.38), (3.98,-2.02)}

(a) Find the circle of best fit for the data.  Do not assume the center is at the origin.
(b) Plot the data and the circle of best fit together on the same graph.

5. Consider the following system of equations:

({{2x+3y==1}, {2x+2y==2}, {-3x+4y==1}, {-3x+3y==2}})

(a) Find the point in the xy-plane which is the closest to being a solution to the system.
(b) Plot the lines and the solution on the same graph.

6. Consider the set of all continuous functions on the interval [0, 2π], denoted by C[0, 2π], and let 

B = {(1/(Sqrt[2π])),(1/(Sqrt[π]))cos(x), (1/(Sqrt[π]))sin(x),(1/(Sqrt[π]))cos(2x),(1/(Sqrt[π]))sin(2x),... , (1/(Sqrt[π]))cos(20x),(1/(Sqrt[π]))sin(20x)}. 

Furthermore, define the dot product of two functions (vectors) in C[0, 2π] by:

f(x)· g(x) = (∫_0^(2π))f(x)*g(x)dx

(a) Show B is an othonormal set.
(b) Project the function F(x) = tanh(3x-1) onto the subspace of C[0, 2π] spanned by B.
(c) Graph F(x) and the projection from (b) in the same graph.

7. Consider the following set of 81 points:

{{-4.1,-3.9,-83.8},{-4.2,-3.9,-49.1},{-3.9,-2.2,-28.3},{-4.1,-1.3,-21.1},{-4.2,0.1,-28.2},{-3.8,0.9,-49.3},{-4.1,1.7,-83.8},{-4.1,3.1,-133.9},{-4.1,4.2,-196.1},{-3.2,-3.7,-86.8},{-3.1,-3.2,-47.4},{-3.1,-2.2,-21.9},{-2.7,-1.2,-12.2},{-3.1,0.4,-14.7},{-2.8,0.7,-31.8},{-2.7,1.7,-62.9},{-3.1,3.5,-109.1},{-3.1,4.1,-167.3},{-2.2,-3.8,-93.7},{-2.1,-2.8,-50.3},{-2.1,-1.8,-21.7},{-2.1,-1.3,-7.1},{-2.2,0.2,-5.8},{-2.1,0.8,-18.7},{-2.1,2.2,-44.9},{-2.1,3.1,-86.7},{-2.1,4.1,-141.1},{-1.4,-3.8,-105.1},{-1.1,-2.8,-57.4},{-1.1,-1.8,-24.7},{-1.1,-1.3,-4.8},{-1.0,0.2,-1.3},{-1.1,1.4,-10.1},{-1.2,2.5,-34.6},{-1.1,3.2,-71.6},{-1.1,4.2,-121.2},{0.1,-4.1,-120.9},{-0.4,-2.8,-68.3},{0.1,-2.1,-31.7},{0.1,-1.3,-8.7},{0.2,-0.2,0.3},{0.2,1.4,-5.2},{0.2,2.1,-23.7},{0.2,3.1,-56.7},{0.1,3.7,-103.8},{1.2,-4.2,-138.7},{1.5,-3.2,-83.4},{1.5,-2.5,-42.7},{1.2,-1.2,-16.4},{1.2,0.1,-3.8},{1.6,1.7,-4.2},{1.1,2.4,-18.2},{1.3,2.6,-47.4},{1.6,3.6,-90.7},{2.1,-3.9,-161.8},{2.7,-2.9,-101.9},{2.1,-2.1,-58.1},{2.2,-1.3,-27.5},{1.9,-0.2,-9.6},{2.6,1.1,-7.1},{2.2,2.3,-18.9},{1.9,3.1,-42.8},{2.2,4.1,-82.2},{3.1,-4.6,-189.1},{3.5,-3.1,-126.1},{3.2,-2.1,-77.1},{3.2,-1.1,-42.1},{3.2,-0.2,-21.5},{2.6,1.2,-13.7},{3.1,1.8,-20.7},{3.1,3.5,-42.1},{3.2,4.1,-76.7},{4.2,-3.8,-220.1},{4.6,-3.2,-153.1},{4.1,-1.8,-100.7},{4.1,-1.1,-61.3},{4.1,0.7,-35.1},{4.7,1.2,-25.9},{4.2,2.5,-28.2},{4.7,3.2,-45.7},{4.3,4.5,-75.8}}

(a) Plot these points in (x,y,z) space.
(b) Find the surface of best fit of the form:

z = a(x^2)+bxy+c(y^2)+dx+ey+f

Here, the unknown constants are {a,b,c,d,e,f}.

(c) Plot the points and the solution on one graph.

(d) Compute the error in your approximate solution of best fit and the data.