Question

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 .

Answer #1

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

a) The standard basis is = {(1,0,0),(0,1,0),(0,0,1)}.

Now, (1,0,0) = (1/2)*(2,1,1)+(-1/2)*(1,2,1)+(1/2)*(1,1,0)

(0,1,0) = (-1/2)*(2,1,1)+(1/2)*(1,2,1)+(1/2)*(1,1,0)

(0,0,1) = (1/2)*(2,1,1)+(1/2)*(1,2,1)+(-3/2)*(1,1,0)

Therefore, the transition matrix from the standard basis of
R^{3} to B_{1} is : Q =
.

b) Here is no information about U. So, it can't be written as
linear combination of the basis B_{1}.

Let B = {u1,u2} where u1 = 1 0and u2 = 0 1 and B' ={ v1 v2]
where v1= 2 1 v2= -3 4 be bases for R2 find 1.the transition matrix
from B′ to B 2. the transition matrix from B to B′ 3.[z]B if z =
(3, −5) 4.[z]B′ by using a transition matrix 5. [z]B′ directly,
that is, do not use a transition matrix.

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 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

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.)

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!

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

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.

Question 6
Suppose u and v are vectors in 3–space where u = (u1, u2, u3)
and v = (v1, v2, v3).
Evaluate u × v × u and v × u × u.

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.

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