Math 311 Final Exam

1. Solve the following system of linear equations.

2. Given and .
a) Show that the span(v₁, v₂) a subspace of R^3.

b) Find a vector v₃ such that v₃ Î span(v₁, v₂)^ .

3. Let P = (1, 3, 1) be a point in R³. Find the distance from P to the plane defined by the equation x + y + z = 0.

4. Let

a) Use the Gram-Schmidt method to find an orthonormal basis for the column space of A.

b) Find the QR factorization for A.

c) Find the LU factorization of A. A = LU.

5. Let

a) Compute B = A^TA

b) Find the characteristic polynomial for B.

c) Find the eigenvalues and eigenvectors of B.

d) Find a U and a diagonal matrix L such that B = ULU^-1.

6. According to Kepler's first law, a comet should have an orbit that is either elliptic, parabolic, or hyperbolic, (ignoring the gravitational attractions from other planets). In suitable polar coordinates, the position (r, q) of a comet from the sun satisfies an equation of the form r = b + e(r cos(q)), where b is a constant and e is the eccentricity of the orbit, with for an ellipse, for a parabola and for a hyperbola. Suppose that observations of a newly discovered comet are given in the table below:

Determine the type of orbit and predict where the comet will be when q = 4.6 (radians).

7. If L: R⁴ ® R² is a map defined by:

a) Determine if L is a linear transformation.

b) Find a basis for Ker(L)

c) Find the standard matrix representation for L, L(x) = Ax.

d) Find a basis for N(A).

e) What is the rank of A?

8. Find a 3x3 matrix R that defines a linear map from R³ to R³ that rotates the xy plane by p/2 radians in the clockwise direction and reflects the z-axis about the xy plane.

a) Is R orthogonal?

b) What is Det(R)?

c) If , what is Ax?

9. The spotted owl population in the Willow Creek area of California is modeled by the dynamical system P_k+1 = L P_k where the entries of P_k = (j_k s_k a_k)^T which represent the size of the female population for the ages juvenile, subadult and adult at a time k in years and

a) Use an eigenvalue analysis to determine if the spotted owls will survive in the Willow Creek area. According to this model how many years will it take before the number of adult females in this area is reduced by 1/2.

b) The 0.18 in the matrix comes from the fact that although 60% of the juvenile owls live long enough to leave the nest only 30% of these live through the search to find a new home. Suppose a combination of resource management and environmental laws could change this percentage, what value must replace the 0.18 to insure a stable population of Adult females?

10. Let S = {v₁, v₂, v₃} and T = {w₁, w₂, w₃} be two subsets of R³ given by:

a) Show S and T are bases for R³.

b) Find the transition matrix from T to S.

c) If [x]_T = what is {x}_S?

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