Question

Let u = [-2,1,3,1] and let v = [1,4,0,1]. a. Determine the projector P1 that projects onto the subspace S1 spanned by the vector u. What isthe rank of P1? b. Determine the projector that projects onto the orthogonal complement of S1. c. Determine the projector P2 that projects onto the subspace S2 spanned y the vectors {u,v}. What is the rank of P2? d. Determine an orthogonal projector that projects onto the orthogonal complement of S2. e. Verify that P1 and P2 are idempoent.

Answer #1

Let W be the subspace of R4 spanned by the vectors a = 3e1 − 4e2
and b = e2 + e3 + e4. Find the orthogonal projection of the vector
v = [2, 0, 1, 0] onto W. Then calculate the distance of the point v
from the subspace W.

Find the orthogonal projection of u onto the
subspace of R4 spanned by the vectors
v1, v2 and
v3.
u = (3, 4, 2, 4) ;
v1 = (3, 2, 3, 0),
v2 = (-8, 3, 6, 3),
v3 = (6, 3, -8, 3)
Let (x, y, z, w) denote the
orthogonal projection of u onto the given
subspace. Then, the components of the target orthogonal projection
are

Let u = ⟨1,3⟩ and v = ⟨4,1⟩.
(a) Find an exact expression and a numerical approximation for
the angle between u and v. (b) Find both the projection of u onto v
and the vector component of u orthogonal to v.
(c) Sketch u, v, and the two vectors you found in part
(b).

For parts ( a ) − ( c
) , let u = 〈 2 , 4 , − 1 〉 and v = 〈 4 , − 2 , 1 〉 .
( a ) Find a unit
vector which is orthogonal to both u and v .
( b ) Find the vector
projection of u onto v .
( c ) Find the scalar
projection of u onto v .
For parts ( a ) − (...

Let U and V be subspaces of the vector space W . Recall that U ∩
V is the set of all vectors ⃗v in W that are in both of U or V ,
and that U ∪ V is the set of all vectors ⃗v in W that are in at
least one of U or V
i: Prove: U ∩V is a subspace of W.
ii: Consider the statement: “U ∪ V is a subspace of W...

let x=u+v. y=v find dS, the a vector, and ds^2 for the u,v
coordinate system and show that it is not an orthogonal system

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.

Do the vectors v1 = 1 2 3 ,
v2 = √ 3 √ 3 √ 3 ,
v3 √ 3 √ 5 √ 7 ,
v4 = 1 0 0 form a basis for R 3 ? Why or why not?
(b) Let V ⊂ R 4 be the subspace spanned by the vectors a1 and
a2, where a1 = ...

(a) Do the vectors v1 = 1 2 3 , v2 = √ 3 √ 3 √ 3 , v3=√ 3 √ 5 √
7, v4 = 1 0 0 form a basis for R 3 ? Why or why not?
(b) Let V ⊂ R 4 be the subspace spanned by the vectors a1 and
a2, where a1 = (1 0 −1 0) , a2 = 0 1 0 −1. Find a basis for the
orthogonal complement V ⊥...

1. V is a subspace of inner-product space R3,
generated by vector
u =[2 2 1]T and v
=[ 3 2 2]T.
(a) Find its orthogonal complement space V┴ ;
(b) Find the dimension of space W = V+ V┴;
(c) Find the angle θ between u and
v and also the angle β between u
and normalized x with respect to its 2-norm.
(d) Considering v’ =
av, a is a scaler, show the
angle θ’ between u and...

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