Math 369 HW #11
Due at the beginning of class on Friday, December 5
Reading: Section O.
Problems:
1. Let V be a vector space. Remember that in class we defined an inner product on V to be
* *
Positive h u, ui ≥ 0
* *
*
*
Definite h u, ui = 0 if and only if u = 0
*
* *
* *
* *
Additive h u + v , wi = h u, wi + h v , wi
* *
* *
Homogeneous hλ u, v iλh u, v i
* *
* *
Symmetric h u, v i = h v , ui
Notice that in this definition we only require the inner product to be homogeneous in the first
factor. Prove that any inner product must be homogeneous in the second factor as well.
* *
In other words, for any λ ∈ R and for any u, v ∈ V , show that the above axioms imply that
*
*
* *
h u, λ v i = λh u, v i.
2. Which of the following
functions
legitimate inner products on R3 ? In each of the
 
 define

u1
v1
*
*



following, u = u2 and v = v2  are arbitrary elements of R3 . If the function is an inner
u3
v3
product just say so, if it is not an inner product, give an explicit example where one of the
axioms fails.
* *
(a) h u, v i = u1 v1 + u3 v3 .
* *
(b) h u, v i = u1 v1 − u2 v2 + u3 v3 .
* *
(c) h u, v i = 2u1 v1 + u2 v2 + 4u3 v3 .
* *
(d) h u, v i = u1 v12 + u22 v22 + u23 v32 .
*
3. Let V be an inner product space and suppose W is a subspace of V . Define W ⊥ = { v ∈ V :
* *
*
h v , wi = 0 for all w ∈ W }, which is called the orthogonal complement of W . Prove that W ⊥
is a subspace of V .
 
 
2
4
*
*
3



4. Consider R with the usual dot product and let u = 2 and v = 3. Find the orthogonal
1
5
*
*
*
*
*
decomposition of u with respect to v . In other words, find vectors x and y so that x is parallel
* *
*
to v , y is perpendicular to v , and
*
*
*
u = x + y.
1
 1   1  

√
√

− √12 


3
6
 
  

5. (a) Show that B =  √13  ,  √16  ,  √12  is an orthonormal basis for R3 (remem

 √1

− √26
0
3
ber that orthonormal sets are automatically linearly independent and that R3 is 3dimensional, so it is enough just to show that B is an orthonormal set).
 
1
*
(b) Write the vector u = 2 as a linear combination of the basis vectors. Equivalently,
3
*
determine [ u]B .
2