Download M340L Exam 1A - Population Dynamics and Linear Algebra - Prof. Arlo W. Schurle and more Exams Mathematics in PDF only on Docsity!
M340L EXAM 1A Your name: SPRING, 2014 Dr. Schurle Your UTEID:
Show all your work on these pages. Be organized and neat. Your work should be your own; there should be no talking, reading notes, checking laptops, using cellphones,....
- (16 points) In a certain region, about 8% of a city’s population moves to the surrounding suburbs each year, and about 6% of the suburban population moves into the city. In 2010, there were 800,000 residents in the city and 500,000 in the suburbs. Set up a difference equation that describes this situation, where x 0 is the initial population in
- Then estimate the populations in the city and in the suburbs three years later, in 2013.
YOUR SCORE: /
- (20 points) State whether each of the following statements is true (T) or false (F). If the given statement is false, then give a true statement as similar as possible to the given one.
(a) Every elementary row operation is reversible.
(b) If every column of an augmented matrix contains a pivot, then the corresponding system is consistent.
(c) When u and v are nonzero vectors, Span{u, v} contains only the line through u and the origin, and the line through v and the origin.
(d) If the columns of an m × n matrix A span Rm, then the equation Ax = b is consistent for each b in Rm.
(e) Every matrix equation Ax = b corresponds to a vector equation with the same solution set.
(f) A homogeneous system of equations can be inconsistent.
(g) The columns of any 4 × 5 matrix are linearly dependent.
(h) If a set in Rn^ is linearly dependent, then the set contains more than n vectors.
(i) If A is an m × n matrix, then the range of the transformation x → Ax is Rm.
(j) A linear transformation T : Rn^ → Rm^ always maps the origin of Rn^ to the origin of Rm.
- (16 points) For which value(s) of h will the following vectors be linearly independent?
,
,
h 3
- (16 points) Do the columns of the following matrix span R^4? Justify your answer.
- (16 points) Suppose the linear transformation T from R^2 to R^3 maps u =
[ 5 3
] into
and v =
[ 8 5
] into
(a) Find T (3u + 5v)
(b) Show that Span{u, v} = R^2.
(c) Use parametric vector form to describe the range of T. Explain why your answer is correct.
