MATH 645 Exam I

(a) (2 points) Calculate the dot products u · v and v · w or explain why they do not exist.
(b) (3 points) Calculate the norms kuk and kvk of u and v.
(c) (2 points) Using your answers to parts (a) and (b) above, confirm that Schwarz’s Inequality holds
for u and v (in other words, verify that |u · v| ≤ kuk · kvk).

(a) (3 points) Are u, v, and w linearly independent or linearly dependent? Justify/explain your response.
(b) (3 points) Consider the set of all linear combinations of u and v. In R
3
, this set of vectors defines
what type of geometric object?

10. (Bonus, 3 points) A band matrix is a square matrix with nonzero entries only on the main diagonal and
on w of the diagonals above and below the main diagonal, and zeros everywhere else. The matrix B
below is an example of a 5×5 symmetric band matrix with w = 1. Elimination for band matrices is
much cheaper than elimination for ordinary matrices. For a generic n × n band matrix B with exactly
w nonzero bands above and below the main diagonal, approximately how many multiplication and
subtraction operations are necessary for elimination B → U?

DETAILED ASSIGNMENT

20200928074536basic_algebra