Question

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.

Answer #1

A). We have u_{1}. u_{2} = 7*(-1)+1*1+4*(-2) =
-7+1-8 = -14. Hence u_{1} and u_{2} are
**not** orthogonal. Further, the vector whose
projection on W is to be computed has not been described. If v =
(a,b,c) is the vector whose projection on W is to be computed, then
proj _{W} (v) = proj _{u1} (v)+ proj _{u2}
(v) =
[(v.u_{1})/(u_{1}.u_{1})]u_{1}
+[(v.u_{2})/(u_{2}.u_{2})]u_{2} = [
(7a+b+4c)/(49+1+16)] (7,1,4)+ [(-a+b-2c)/(1+1+4)] (-1,1,-2) =
[(7a+b+4c)/66] (7,1,4) + [(-a+b-2c)/6](-1,1,-2)=(49a+7b+28c)/66,
(7a+b+4c)/66, (28a+4b+16c)/66)+((a-b+2c)/6, (-a+b-2c)/6,
(2a-2b+4c)/6) = (
(30a-2cb+25c)/33,(-2a+6b-9c)/33,(25a-9b+30c)/33).

You have to plugin the values of a, b, c.

B).We have u_{1}= (1,1,1), u_{2}= (1/3,1/3,-2/3)
and W=span{u_{1},u_{2}}. Since
u_{1}.u_{2} = 1/3+1/3-2/3 = 0, hence
u_{1},u_{2} are orthogonal to each other.

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.

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.

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.

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.

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)

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.)

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.

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!

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.

Let B={u1,...un} be an orthonormal basis for inner product space
V and v b any vector in V. Prove that v =c1u1 + c2u2 +....+cnun
where c1=<v,u1>, c2=<v,u2>,...,cn=<v,un>

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