Question

(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).

Answer #1

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 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.

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 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 = ___________

What are the resulting orthonormal vectors after applying the
Gram-Schmidt process to the 3x1 vectors: [3 1 -2], [2 -1 -1], [1 1
2]?

a) Apply the Gram–Schmidt process to find an orthogonal basis
for S.
S=span{[110−1],[1301],[4220]}
b) Find projSu.
S = subspace in Exercise 14; u=[1010]
c) Find an orthonormal basis for S.
S= subspace in Exercise 14.

Use the Gram-Schmidt process to construct the first four
orthonormal polynomials for the following intervals and weights
(a) w(x) = 1, [1, 2]
ϕ0x=1

convert the basis V1=(1,-1,0), v2=(0,1,-1), v3=(-1,1,-1)for R^3
into an orthonormal basis, using theGram-Schmidt process and the
standard inner product in R^3

Apply the Gram-Schmidt process to the vectors [1; −2; 0], [1; 0;
−1] and [0; 1; 1] .

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