Question

Use the Gram-Schmidt process to find an orthonormal basis for
the subspace of *R*^{4} spanned by the vectors

u_{1} = (1, 0, 0, 0), u_{2} = (1, 1, 0, 0),
u_{3} = (0, 1, 1, 1).

Show all your work.

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.

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 vectors
u1 = (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.

Use Gram-Schmidt process to transform the basis {u1, u2, u3},
where u1=(1,1,1), u2=(1,2,0), u3=(1,0,-1),:
a) for the Euclidean IPS. (IPS means inner product space)

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 =

Apply the Gram-Schmidt orthonormalization process to transform
the given basis for into an orthonormal basis„ Use the vectors in
the order in which they are given.
B={1,1,1>,<-1,1,0>,<1,2,1>

Apply the Gram-Schmidt orthonormalization process to transform
the given basis for
ℝ4 into an orthonormal basis. Use the vectors in the
order in which they are given.
B ={(3,4,0,0),(−1,1,0,0),(2,1,0,−1),{0,1,1,0}}

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 = ___________

If W is a subspace of Rn with an orthonormal basis
u1, u2, . . . , uk, if x ∈
Rn, and if
projW (x) = (x • u1)u1 + (x •
u2)u2 + · · · + (x •
uk)uk,
then x − projW (x) is orthogonal to every element of
W.
(Please show that x − projW (x) is orthogonal to each
uj for 1 ≤ j ≤ k)
u,x are vectors.

Let u1= [4,2,2], u2=[20,12,8], and
u3=[-20,66,58]. Apply the Gram-Schmidt orthogonalization
process and find w3.

(a) Use the Gram-Schmidt process on the basis {(1, 2, 2),(1, 2,
3),(4, 3, 2)} of R ^3 find an orthonormal basis.
(b) Write the vector v = (2, 1, −5) as a linear combination of
the orthonormal basis vectors found in part (a).

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