Study with the several resources on Docsity
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Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
This is the Past Exam of Linear Algebra which includes Row Equivalent, Scalars, Column Vectors, Components, Values, System of Equations, Matrix Equation, Represented, Special Conditions etc. Key important points are: Polynomials, Forms, Basis, Coordinate Vector, Relative, Basis, Subspace, Dimension, Vectors, Compute
Typology: Exams
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While the final answer is important, you earn points for all the work leading to that answer, as well as the answer itself. Show all your steps clearly so you will be eligible for the most partial credit. Good luck!
1.) (15 pts.) Determine whether the set of polynomials {3 + 7t, 5 + t − 2 t^3 , t − 2 t^2 , 1 + 16t − 6 t^2 + 2t^3 } forms a basis for P 3. Justify your conclusion.
2.) (10 pts.) Find the coordinate vector [x]B of x =
(^) relative to the basis B =
3.) (10 pts.) For the subspace
s − 2 t s + t 3 t
(^) : s, t in R
a.) find a basis, and
b.) state the dimension.
5.) (10 pts.) Find the characteristic polynomial and the eigenvalues of the matrix
6.) (10 pts.) Let A be an n × n matrix. True or False: An eigenspace of A is a null space of a certain matrix. If True: explain why in detail. If False: explain why in detail, and/or provide a counterexample, that is, an example to show when the statement is false.
7.) (15 pts.) A scientist solves a nonhomogeneous system of ten linear equations in twelve unknowns and finds that three of the unknowns are free variables. Can the scientist be certain that, if the right sides of the equations are changed, the new nonhomogeneous system will have a solution? Discuss.
