Question

3. a. Consider R^2 with the Euclidean inner product (i.e. dot product). Let

v = (x1, x2) ? R^2. Show that (x2, ?x1) is orthogonal to v.

b. Find all vectors (x, y, z) ? R^3 that are orthogonal (with the Euclidean

inner product, i.e. dot product) to both (1, 3, ?2) and (2, 7, 5).

C.Let V be an inner product space. Suppose u is orthogonal to both v

and w. Prove that for any scalars c and d, u is orthogonal to cv + dw.

Answer #1

Chapter 6, Section 6.2, Question 16
Let R4 have the Euclidean inner product.
Find two unit vectors that are orthogonal to all three vectors
u = (4, 3, -8, 0), v = (-1, -1,
2, 2), and w = (3, 2, 5, 5). Give the exact
answers in increasing order of the first component.
X1?
(__,__,__,__)
X2?
(__,__,__,__)

A2. Let v be a fixed vector in an inner product space V. Let W
be the subset of V consisting of all vectors in V that are
orthogonal to v. In set language, W = { w LaTeX: \in
∈V: <w, v> = 0}. Show that W is a subspace of V. Then,
if V = R3, v = (1, 1, 1), and the inner product is the usual dot
product, find a basis for W.

Let W be an inner product space and v1,...,vn a basis of V. Show
that〈S, T 〉 = 〈Sv1, T v1〉 + . . . + 〈Svn, T vn〉
for S,T ∈ L(V,W) is an inner product on L(V,W).
Let S ∈ L(R^2) be given by S(x1, x2) = (x1 + x2, x2) and let I ∈
L(R^2) be the identity operator. Using the inner product defined in
problem 1 for the standard basis and the dot product, compute 〈S,...

Let V be an inner product space. Prove that if w⃗ is orthogonal
to each of the vectors in the set
S = {⃗v1, ⃗v2, . . . , ⃗vm}, then w⃗ is also orthogonal to each
of the vectors in the subspace W = SpanS of V .

let v be an inner product space with an inner product(u,v) prove
that ||u+v||<=||u||+||v||, ||w||^2=(w,w) , for all u,v load to
V. hint : you may use the Cauchy-Schwars inquality: |{u,v}|,=
||u||*||v||.

Suppose that u and v are two non-orthogonal vectors in an inner
product space V,< , >.
Question 2: Can we modify the inner product < , > to a new
inner product so that the two vectors become orthogonal? Justify
your answer.

Let S ∈ L(R2) be given by S(x1, x2) = (x1 +x2, x2) and let I ∈
L(R2) be the identity operator.
Using the inner product defined in problem 1 for the standard basis
and the dot product,
compute <S, I>, || S ||, and || I ||
{Inner product in problem 1: Let W be an inner product space and
v1, . . . , vn a basis of V. Show that <S, T> = <Sv1, T
v1> +...

Suppose V is a ﬁnite dimensional inner product space. Prove that
every orthogonal operator on V , i.e. <T(u), T(v)> , ∀u,v ∈ V
, is an isomorphism.

Let W be a subspace of a f.d. inner product space V and let PW
be the orthogonal projection of V onto W. Show that the
characteristic polynomial of PW is
(t-1)^dimW t^(dimv-dimw)

Let V be the vector space of 2 × 2 matrices over R, let <A,
B>= tr(ABT ) be an inner product on V , and let U ⊆ V
be the subspace of symmetric 2 × 2 matrices. Compute the orthogonal
projection of the matrix A = (1 2
3 4)
on U, and compute the minimal distance between A and an element
of U.
Hint: Use the basis 1 0 0 0
0 0 0 1
0 1...

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