Mathematics 44, section 1 -- Linear Algebra
Discussion 4 -- Another Dimension Formula
February 26, 1999
Background
In class on Wednesday, we proved the following
Dimension Theorem. Let T: V -> W be a linear mapping
and let V be finite-dimensional. Then
dim(V) = dim(Ker(T)) + dim(Im(T)).
Today, we want to use this result (and/or other facts) to derive another
dimension formula relating the dimension of the sum of two subspaces
U,W of a vector space V, dim(U + W) to the dimensions
dim(U) and dim(V).
Discussion Questions
- A) Let V = W = P4(R) and consider the mapping
T: V -> W defined by
T(p(x)) = x2p''(x) - 3xp'(x) + 3p(x).
Determine bases for Ker(T) and Im(T)
and show that the conclusion
of the Dimension Theorem is satisfied.
- B) First, we need to have a good idea of what
we want to try to prove! In mathematics, a plausible hunch or guess
about what is going on in a particular situation or problem
is often called a conjecture. So, we
need first to examine some evidence and make a conjecture that matches
what we see in some examples. Here are two instructive ones:
- V = R4,
U = L({(1,1,2,-1),(-1,0,-1,2)}) and
W = L({(1,3,4,1),(0,1,-2,6)}). What are
dim(U), dim(W)? How can you get a basis for U + W
to determine dim(U + W)? What is dim(U+W)?
- V = R4 again,
U = L({(1,1,2,-1),(-1,0,-1,2)}), W' = L({(1,3,4,2),(0,1,-2,6)}).
Same questions as in 1.
(Note: U is the same as
in part 1. In the spanning set for W', only one component has
been changed in the first vector.)
- Why is 1 different from 2?
- Is there a general conjecture you can make that
matches what you see in these examples? Make up another example
and see if your conjecture holds there too.
- The next few questions are designed to ``lead'' you to one proof of a
general formula for dim(U + W). If you have a different idea
for a proof of your conjecture from A, that's OK too, but you're
on your own then and you'll need to develop a different outline for
the proof!
- C) As in problems from the first problem set, given the
vector spaces U and W we can form the Cartesian product
space
U x W = {(A,B) : A in U, B in W}
If you know finite bases for U and W, how can you get a basis
for U x W? What is dim(U x W)?
(Be careful, this construction takes you outside of the vector
space V; U x W is not a subset of V!)
- D) Consider the mapping:
T : U x W -> V defined by T(A,B) = A + B.
What are Im(T), Ker(T)?
- E) What does the Dimension Theorem applied to the mapping
T from part C tell you?
Assignment
Group write-ups due Wednesday, March 3.