Question

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.

Answer #1

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!

A): compute projw j if u1=[-7,1,4] u2=[-1,1,-2],w=span{u1,u2}.
(u1 and u2 are orthogonal)
B): let u1=[1,1,1], u2=1/3 *[1,1,-2] and w=span{u1,u2}.
Construct an orthonormal basis for w.

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)

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

write the vector w=(1,-4,13) as a linear combination of u1=(1,2,3),
u2=(2,1,1), u3=(1,-1,2)

Let B1 = { u1, u2, u3 }, where u1 = (2,?1, 1), u2 = (1,?2, 1),
and u3 = (1,?1, 0). B1 is a basis for R^3 .
A. Find the transition matrix Q ^?1 from the standard basis of R
^3 to B1 .
B. Write U as a linear combination of the basis B1 .

If v1 and v2 are linearly independent vectors in vector space V,
and u1, u2, and u3 are each a linear combination of them, prove
that {u1, u2, u3} is linearly dependent.
Do NOT use the theorem which states, " If S = { v 1 , v 2 , . . . ,
v n } is a basis for a vector space V, then every set
containing
more than n vectors in V is linearly dependent."
Prove without...

SSTR=6750
SSE= 6000
Nt =20
Ho: u1=u2=u3=u4
Ha: at least one mean is different
At Alpha=.05 the null hypothesis:
A should be rejected
B should not be rejected
C was designed incorrectly
D none of the above

Here are some vectors in R 4 : u1 = [1 3 −1 1] u2 = [1 4 −1 1]
u3 = [1 0 −1 1] u4 = [2 −1 −2 2] u5 = [1 4 0 1]
(a) Explain why these vectors cannot possibly be
independent.
(b) Form a matrix A whose columns are the ui’s and compute the
rref(A).
(c) Solve the homogeneous system Ax = 0 in parametric form and
then in vector form. (Be sure the...

4. Prove the Following:
a. Prove that if V is a vector space with subspace W ⊂ V, and if
U ⊂ W is a subspace of the vector space W, then U is also a
subspace of V
b. Given span of a finite collection of vectors {v1, . . . , vn}
⊂ V as follows:
Span(v1, . . . , vn) := {a1v1 + · · · + anvn : ai are scalars in
the scalar field}...

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