Newberger Math 247 Spring 03
Writing Assignment Suggestions For “Linear Independence” Here are
some ideas about which you may write for the writing assignment due Mon-
day February 25th. The assignment is “Write about linear independence” (350
words). You may use any or none of these ideas.
•Relate the linear independence of a set of vectors {v1, . . . , vn}to the span
of that set of vectors, Span{v1, . . . , vn}.
•Explain why saying the matrix equation [v1, . . . , vn]x=0has nontrivial
solutions means one of the vectors vican be written in terms of the other
vectors in the list.
•Explain why having a pivot in every column in the matrix [v1, . . . , vn]
means none of the vectors can be linear combinations of the others.
•Use linear independence to understand (better than in the last assignment)
the span of three vectors in R3.
•Look at the Theorems in Section 1.7; can you explain why they are true?
What examples work well to illustrate the concepts in the theorems. Why
do they work well?
•If you are planning to teach math, you might be interested in thinking
about how to best explain mathematical concepts. Look at the author’s
use of examples to illustrate the concepts in the section on linear indepen-
dence. How does the author use the examples pedagogically? To warm
up the reader before explaining a difficult concept? To reinforce ideas
from previous chapters? Focus on the concepts and what is difficult about
them, and how the author tries to combat that difficulty. (Make sure you
discuss linear independence in particular for this assignment!)
•Challenging idea: When you write a solution set of a homogeneous equa-
tion Ax=0in parametric vector form, are the vectors that show up in
your solution always linearly independent? Suggestion: try some examples
by making up a homogeneous equation, solving it, putting the solution in
parametric vector form and looking at the vectors that you get.
It might be a good idea to start with the definition of a linearly independent
set. Expand on your ideas by making up examples of the concepts you are
discussing. If you notice a pattern in what is occurring in your examples, try to
explain that pattern. Maybe invent a Theorem based on your observations and
explain why it is correct?