Page 362 - A Course in Linear Algebra with Applications

346 Chapter Nine: Advanced Topics

Exercises 9.3

1. Which of the following functions / are bilinear forms?
n
(a) f(X,Y) = X~Y on R -
n
T
(b) f(X,Y) =X Y onR ;
(c) f(g,h) = f ag{x)h(x) dx on C{a,b].
2. Let / be the bilinear form on R 2 which is defined by the
equation f(X,Y) = 2x\y2 — 3x2yi- Write down the matrices
which represent / with respect to (a) the standard basis, and
(b) the basis {( J J ( j }.
)

3. If / and g are two bilinear forms on a vector space V, define
their sum / + g by the rule / -f- g(u,v) = f(u,v) + g(u,v);
also define the scalar multiple cf by the equation cf(u, v) =
c(f(u,v)). Prove that with these operations the set of all
bilinear forms on V becomes a vector space V . If V has
dimension n, what is the dimension of V ?

4. Prove that every bilinear form on a real or complex vector
space is the sum of a symmetric and a skew-symmetric bilinear
form.

5. Find the canonical form of the symmetric bilinear form on
R 2 given by f(X, Y) = 3ziyi + x xy 2 + x 2yi + 3x 2y2-
n
6. Let / be a bilinear form on R . Prove that / is an inner
product on R n if and only if / is symmetric and the corre-
sponding quadratic form is positive definite.
7. Test each of the following bilinear forms to see if it is an
inner product:
(a) f(X, Y) = 3zi2/i + x xy 2 + x 2yi + 5x 2y2]
(b) f(X,Y) = 2x 1y 1+xiy2+x 1y 3 + x 2yi + 3x 2y2-2x2y3
+x 3yt - 2x 3y2 + 3x 3y 3.