Question

Let
R4
have the inner product< u, v> =
u1v1 +
2u2v2 +
3u3v3 +
4u4v4 |

(a) | Let w = (0, 6,
4, 1). Find ||w||. |

(b) | Let W be the
subspace spanned by the vectorsu1 = (0, 0, 2,
1), and u2 = (3, 0, −2,
1).Use the Gram-Schmidt process to transform the basis { u1,
u2} into an
orthonormal basis {v1,
v2}. Enter the
components of the vector v2 into the
answer box below, separated with commas. |

Answer #1

Let W be the subspace of R4
spanned by the vectors
u1 = (−1, 0, 1, 0),
u2 = (0, 1, 1, 0), and
u3 = (0, 0, 1, 1).
Use the Gram-Schmidt process to transform the basis
{u1, u2,
u3} into an orthonormal basis.

Use the inner product (u, v) =
2u1v1 +
u2v2 in
R2 and the Gram-Schmidt orthonormalization
process to transform {(?2, 1), (2, 5)} into an orthonormal basis.
(Use the vectors in the order in which they are given.)
u1 = ___________
u2 = ___________

Use the inner product <u,v>= 2u1v1 + u2v2 in R2 and the
Gram-Schmidt orthonormalization process to transform {(2, ?1), (2,
6)} into an orthonormal basis. (Use the vectors in the order in
which they are given.)
u1 =
u2 =

Use the Gram-Schmidt process to find an orthonormal basis for
the subspace of R4 spanned by the vectors
u1 = (1, 0, 0, 0), u2 = (1, 1, 0, 0),
u3 = (0, 1, 1, 1).
Show all your work.

(i) Let u= (u1,u2) and v=
(v1,v2). Show that the following is an inner
product by verifying that the inner product hold
<u,v>= 4u1v1 +
u2v2 +4u2v2
(ii) Let u= (u1,
u2, u3) and v=
(v1,v2,v3). Show that the
following is an inner product by verifying that the inner product
hold
<u,v> =
2u1v1 + u2v2 +
4u3v3

Find the orthogonal projection of u onto the
subspace of R4 spanned by the vectors
v1, v2 and
v3.
u = (3, 4, 2, 4) ;
v1 = (3, 2, 3, 0),
v2 = (-8, 3, 6, 3),
v3 = (6, 3, -8, 3)
Let (x, y, z, w) denote the
orthogonal projection of u onto the given
subspace. Then, the components of the target orthogonal projection
are

Let W be the subspace of R4 spanned by the vectors a = 3e1 − 4e2
and b = e2 + e3 + e4. Find the orthogonal projection of the vector
v = [2, 0, 1, 0] onto W. Then calculate the distance of the point v
from the subspace W.

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 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 the set of all ordered pairs of real numbers. Consider
the following addition and scalar multiplication operations V. Let
u = (u1, u2) and v = (v1, v2).
• u ⊕ v = (u1 + v1 + 1, u2 + v2 + )
• ku = (ku1 + k − 1, ku2 + k − 1)
1)Show that the zero vector is 0 = (−1, −1).
2)Find the additive inverse −u for u = (u1, u2). Note:...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 4 minutes ago

asked 19 minutes ago

asked 19 minutes ago

asked 22 minutes ago

asked 37 minutes ago

asked 40 minutes ago

asked 40 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago