Problem Set 5 (due Friday October 8)
MATH 110: Linear Algebra
Each part of each problem is worth 5 points.
PART 1
1. Curtis p. 129 1.
2. Curtis p. 130 11.
Remember that the starred problem is non collaborative.
Problem 1(10)
An inner product on a vector space Vis a function which assigns to each
pair of vectors u, v in Va real number such that the following conditions are
satisfied (cis any real number, wis any vector in V):
1. (u, v) = (v, u).
2. (u, v +w) = (u, v)+(u, w).
3. c(u, v) = (cu, v).
4. (u, u)>0 if u6= 0.
Show that this definition of an inner product is equivalent to the ax-
iomatic definition given in Curtis, page 119.
Problem 2(10)
Let C(R) be the vector space of all polynomials with inner product
(x, y) = Z1
โ1
x(t)y(t)dt.
Consider the subspace of C(R) generated by the polynomials {1, x, x2, x3, x4}.
Use the Gram-Schmidt process to find polynomials y0, y1, y2, y3, y4that are
orthonormal and span the same subspace.
Problem 3โ(10)
Let Ube a subspace of a vector space V. Show that if 0 is the only vector
orthogonal to all the vectors in Uthen U=V.
PART 3 - Optional Problem
a) Let Sbe a finite dimensional subspace of a Euclidean space Vand let
e1, e2, . . . , enbe an orthonormal basis for S. For any xโV, let
s=
n
X
i=1
(x, ei)ei
1
