Question

Let u_{1}= [4,2,2], u_{2}=[20,12,8], and
u_{3}=[-20,66,58]. Apply the Gram-Schmidt orthogonalization
process and find w_{3}.

Answer #1

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)

5. Let U1, U2, U3 be subspaces
of a vector space V.
Prove that U1, U2, U3 are
direct-summable if and only if
(i) the intersection of U1 and U2 is 0.\,
and
(ii) the intersection of U1+U2 and
U3 is 0.
A detailed explanation would be greatly appreciated :)

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.

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.

Consider the vector u1=(2,0,2), u2=(4,1,-1), u3=( 0,1,-5),
u4=(3,0,2)
a) Find the dimension and a basis for U= span{ u1,u2,u3,u4}
b) Does the vector u=(2,-1,4) belong to U. Justify!
c) Is it true that U = span{ u1,u2,u3} justify the answer!

Determine if the vector v is a linear combination of the vectors
u1, u2, u3. If yes, indicate at least one possible value for the
weights. If not, explain why.
v =
2
4
2
, u1 =
1
1
0
, u2 =
0
1
-1
, u3 =
1
2
-1

Vectors u1= [1,1,1] and u2=[8,-7,-1] are
perpendicular. Find the orthogonal projection of
u3=[65,-19,-31] onto the plane spanned by u1
and 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 =

Consider the vector: u1=(1,1,1), u2= (2,-1,1), u3=(3,0,2),
u4=(6,0,4)
a)Plot the dimension and a basis for W= span {u1,u2,u3,u4}
b)Does the vector v= (3,3,1) belong to W? justify the
answer.
c) Is it true that W= span{ u3,u4} Justify answer.

Linear Algebra
Write x as the sum of two vectors, one is Span {u1,
u2, u3} and one in Span {u4}. Assume that
{u1,...,u4} is an orthogonal basis for
R4
u1 = [0, 1, -6, -1] , u2 = [5, 7, 1, 1],
u3 = [1, 0, 1, -6], u4 = [7, -5, -1, 1], x =
[14, -9, 4, 0]
x =
(Type an integer or simplified fraction for each matrix
element.)

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