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September 27 Mathematics 205 Mr. Haines 2004 Linear Algebra Examination #
(10) I. Suppose b =
and the columns of the matrix A are
, and
A. Determine whether or not the equation A x = b has a solution and explain your reasoning.
B. Determine whether or not b is in the span of the columns of A and explain your reasoning.
(10) II. Give an example of a linear system of equations in two variables whose solution set is a straight line.
(5) III. Give a parametric vector equation of the line through the
point
parallel to the vector
(10) VI. Suppose an economy has two sectors, Goods and Services. Each year, Goods sells 40% of its output to services and keeps the rest, while Services sells 70% of its output to Goods and keeps the rest. Find equilibrium prices for the annual outputs of the Goods and Services sectors that make each sector’s income match its expenditures.
(10) VII. If T : ℜ 2 →ℜ^2 reflects points through the line x 1 = x 2 , give the standard matrix of the linear transformation T.
(10) VIII. The augmented matrix of a linear system has been reduced by row operations to the form
Reduce this augmented matrix reduced row echelon form and describe the solution set of the original system as a parametric vector equation.
(5) IX. Suppose T( x ) = A x , where A =
Without reducing this matrix to reduced echelon form, explain why T is not one-to-one.
