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M340L EXAM 1A 1:00 Your name: FALL, 2012 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) Consider the task of pulling a weight of 40 dan (A dan was about 30 kg at the time this problem was written.) We have one military horse, three ordinary horses, and four weak horses at our disposal to get the job done. It turns out that the military horse and one of the ordinary horses, pulling together, are barely able to pull the weight (but they could not pull any more). Likewise, the three ordinary horses together with one weak horse are just able to do the job, as are the four weak horses together with the military horse. How much weight can each of the horses pull alone? You may leave your answers in dans. [This problem comes from a Chinese text, Nine Chapters on the Mathematical Art, which is more than 2,000 years old.]
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.
