Mathematics 44, section 1 -- Linear Algebra
Discussion 3 -- Mappings Between Vector Spaces
February 15, 1999
Background
In Algebraic Structures, in our study of groups, we introduced
the idea of a group homomorphism. Recall that a group
homomorphism was a mapping from one group G to another
group H, f : G -> H, with the property that
f(g *G h) = f(g) *H f(h)
for all g,h in G. In
other words, f ``preserves the structure of the group G''
in the sense that the image of the product g *G h in
G is the
same as the product of the images f(g) and f(h) in H.
Today, we want to begin to study the corresponding mappings for
vector spaces -- the mappings T from one vector
space V to another vector space W, T : V -> W, that
``preserve the structure of the vector space V'' in an analogous
way.
Discussion Questions
- A) First, just what should it mean for a mapping T : V -> W
to ``preserve the structure of the vector space V''? Give a
proposal for a definition of what this should mean, and explain your
reasoning. Recall that the definition of a vector space includes
both the vector sum and the scalar multiplication operations.
- B) Which of the following mappings satisfy your proposed
definition?
- V = W = R2,
and T : R2 -> R2
defined by
T(x,y) = (3x+5y,-x + y).
- V = R3, W = R2, and
T : R3 -> R2 defined by
vertical projection on the xy-plane:
T(x,y,z) = (x,y,0).
- V = W = R2, and T: V -> W defined
by T(x,y) = (x2 - y2, 2xy - 3).
- V = P3(R), W = P4(R),
and T : P3(R) -> P4(R)
defined by T(p(x)) = int0x p(t) dt
- V = D = { f in Fun(R) : f is
differentiable for all x in R},
W = Fun(R), and T : V -> W defined
by T(f) = f'(x) - sin(x)f(x) - cos(x).
- C) Suppose that T : V -> W does preserve the structure
of the vector space V according to your definition. Assume
that E = {A1, ... , An} is a
basis for V. If
you know the vectors T(A1), ... ,T(An)
in W, can you
determine T(A) for all A in V?
Why or why not? If so, say how you can do this; if not, say why not.
Assignment
Group write-ups due Monday, February 22.