Question

Find the orthogonal projection of **u** onto the
subspace of *R*^{4} spanned by the vectors
**v**_{1}, **v**_{2} and
**v**_{3}.

**u** = (3, 4, 2, 4) ;
**v**_{1} = (3, 2, 3, 0),
**v**_{2} = (-8, 3, 6, 3),
**v**_{3} = (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

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.

let v1=[1,0,10], v2=[0,1,0,1] and let W be the
subspace of R^4 spanned by v1 and v2.
A. convert {v1,v2} into an orhonormal basis of W.
Basis =
B.find the projection of b=[-1,-2,-2,-1] onto W
C.find two linear independent vectors in R^4
perpendicular to W.
vectors =

Find the orthogonal projection of v⃗ =⎢4,−11,−36,9⎤ onto the
subspace W spanned by ⎢0,0,−5,−2| , |−4,2,5,−5⎢ , ⎢−5,−5,0,5|

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.

Find the orthogonal projection of v=[−2,10,−16,−19] onto the
subspace W spanned by [-4,0,-2,1],[-4,-2,5,1],[3,-1,-3,4]

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 ⊥...

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.

first use Gram-Schmidt on x1, x2 to create orthogonal vectors v1
and v2 with the same span as x1, x2. Now use the formula
p =((y, v1)/(v1, v1))v1 + ((y, v2)/(v2, v2))v2
to compute the projection of y onto that span. Of course,
replace the inner product with the dot product when working with
standard vectors
1)
Compute the projection of y = (1, 2, 3) onto span
(x1, x2) where
x1 =(1, 1, 1) x2 =(1, 0, 1)
The inner...

Exercise 6. Consider the following vectors in R3 . v1 = (1, −1,
0) v2 = (3, 2, −1) v3 = (3, 5, −2 ) (a) Verify
that the general vector u = (x, y, z) can be written as a linear
combination of v1, v2, and v3. (Hint : The coefficients will be
expressed as functions of the entries x, y and z of u.) Note : This
shows that Span{v1, v2, v3} = R3 . (b) Can R3 be...

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