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This is the Past Exam of Linear Algebra which includes Vectors, Angle, System, Factorization, Matrix Pascal, Matrix, Reflects Vectors, Line Making Angle, Same Line etc. Key important points are: System, Required, Solutions, Unique Solution, In Nitely, Properties, Subspace, Mapping, Linear Transformation, Elements
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Mathematics 205 Exam I February 17, 2012
Problem Possible Actual 1 15 2 15 3 6 4 8 5 12 6 14 7 15 8 15 Total 100
You must show all work to receive credit. No electronic devices other than calculators are permitted. Give exact answers (such as ln 5 or e^2 ) unless requested otherwise.
1 2 k 3 h 8
(a) What is required of h and k so that the system has no solutions?
(b) What is required of h and k so that the system has a unique solution?
(c) What is required of h and k so that the system has infinitely many solutions?
. Write basis elements for nullB and colB.
For example, π + 3x + x^2 , 4 + 8x, and a 0 + a 1 x + a 2 x^2 have the vectors
π 3 1
, and
a 0 a 1 a 2
(a) Let y′(t) be an element of S and define
T (y′) =
∫ (^) x 0 y
′(t) dt x
Show that T is a linear transformation.
(b) What is the matrix of the transformation of this map. (Hint: The vector ~e 2 =
(^) corresponds
to polynomial x. Where is x mapped to and what is the vector that corresponds to that answer?)
PbN 6 + CrMn 2 O 8 → Pb 3 O 4 + Cr 2 O 3 + MnO 2 + NO.
