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The solutions manual for exam 3a of a linear algebra course, including problems on eigenvalues, eigenvectors, orthonormal matrices, and linear transformations.
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M340L EXAM 3A Your name: SPRING, 2010 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,....
an eigenvector of the matrix
? If so, what is
its eigenvalue? If not, explain why not.
(a) (4 points) Calculate and simplify T (2t^2 + 3t − 1).
(b) (8 points) What is the matrix for T relative to the standard basis for P 2?
(c) (8 points) What is the matrix for T relative to the basis B = { 1 , t − 1 , (t − 1)^2 }?
u 1 =
, u 2 =
, u 3 =
.
(a) (4 points) Verify that {u 1 , u 2 , u 3 } is an orthogonal basis for W.
(b) (10 pts.) Find (and simplify) the vector in W that is closest to y =
.
(c) (6 points) Find the distance between y and the vector in W that is closest to y.
(d) (6 points) Find a nonzero vector in W ⊥.
