Download MAT340 Exam 2: Linear Algebra Problems and more Exams Linear Algebra in PDF only on Docsity!
MAT340 Exam 2 Due 14 November, 2006 Prof. Thistleton
You may consult written sources such as textbooks or your class notes as you work through this exam. You may not, however, work with or receive help from anyone on this exam.
- Let
V = {v 1 , v 2 , v 3 } where v 1 =
, v 2 =
, v 3 =
and
U = {u 1 , u 2 , u 3 } where u 1 =
, u 2 =
, u 3 =
.
(a) Express x =
in terms of basis V.
(b) Express x =
in terms of basis^ U^.
(c) Find the matrix S which transforms vectors in terms of basis V to basis U and demon- strate that this works for your vector x.
- Define the linear operator L : <^3 → <^3 as
L(x) = L([x 1 , x 2 , x 3 ]T^ ) = [2x 1 − x 3 , x 2 + x 3 , x 1 − x 2 ]T
Find the matrix representation of this linear transformation with respect to the standard basis and use it to calculate L([1, 1 , 1]T^ ).
- If L : <^2 → <^2 is such that
L([3, −2]T^ ) = [2, 2]T^ and L([2, 1]T^ ) = [3, 5]T
then calculate L([10, −9]T^ ).
(d) Show that B = S−^1 AS.
- Let A =
Find a basis for the row space, the column space and the null space of A.
- Show that the vectors u = ex^ and v = x^2 are independent in the space C(−∞, ∞).
- Are the vectors u = x + 1, v = x − 1 and w = x^2 + x − 1 independent in the space P 3?
