Question

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]

Answer #1

Let v_{1} = [-4,0,-2,1], v_{2} = [-4,-2,5,1] and
v_{3} = [3,-1,-3,4].

We have
proj_{v1}(v)=[(v.v_{1})/(v_{1}.v_{1})]v_{1}=[(
8+0+32-19)/(16+0+4+1)]v_{1}=v_{1}=[-4,0,-2,1],
proj_{v2}(v)=[(v.v_{2})/(v_{2}.v_{2})]v_{2}=[(
8-20-80-19)/(16+4+25+1)]v_{2} = (111/46)[-4,-2,5,1]=
[-222/23, -111/23,555/46,111/46] and
proj_{v3}(v)=[(v.v_{3})/(v_{3}.v_{3})]v_{3}=[(-6-10+48+76)/(9+1+9+16)]v_{3}
= (108/35) [3,-1,-3,4] = [ 324/35,-108/35,-324/35,432/35].

Hence, **proj _{W}(v)=**
proj

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

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 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.

1. Find the orthogonal projection of the matrix
[[3,2][4,5]] onto the space of diagonal 2x2 matrices of the form
lambda?I.
[[4.5,0][0,4.5]] [[5.5,0][0,5.5]] [[4,0][0,4]] [[3.5,0][0,3.5]] [[5,0][0,5]] [[1.5,0][0,1.5]]
2. Find the orthogonal projection of the matrix
[[2,1][2,6]] onto the space of symmetric 2x2 matrices of trace
0.
[[-1,3][3,1]] [[1.5,1][1,-1.5]] [[0,4][4,0]] [[3,3.5][3.5,-3]] [[0,1.5][1.5,0]] [[-2,1.5][1.5,2]] [[0.5,4.5][4.5,-0.5]] [[-1,6][6,1]] [[0,3.5][3.5,0]] [[-1.5,3.5][3.5,1.5]]
3. Find the orthogonal projection of the matrix
[[1,5][1,2]] onto the space of anti-symmetric 2x2
matrices.
[[0,-1] [1,0]] [[0,2] [-2,0]] [[0,-1.5]
[1.5,0]] [[0,2.5] [-2.5,0]] [[0,0]
[0,0]] [[0,-0.5] [0.5,0]] [[0,1] [-1,0]]
[[0,1.5] [-1.5,0]] [[0,-2.5]
[2.5,0]] [[0,0.5] [-0.5,0]]
4. Let p be the orthogonal projection of
u=[40,-9,91]T onto the...

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.

U= [2,-5,-1] V=[3,2,-3] Find the orthogonal projection of u onto
v. Then write u as the sum of two orthogonal vectors, one in
span{U} and one orthogonal to U

Linear Algebra:
Find the orthogonal projection of u3=[48,-12,108] onto the plane
spanned by u1= [2,7,2] and u2=[5,35,15].
Answer Choices: [15,3,-3] [35,23,-34] [5,1,3] [-22,28,38]
[24,21,-6] [24,0,6] [34,14,18] [21,22,-11] [57,12,27]
[39,37,15]

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 =

Let W be a subspace of a f.d. inner product space V and let PW
be the orthogonal projection of V onto W. Show that the
characteristic polynomial of PW is
(t-1)^dimW t^(dimv-dimw)

Let W be a subspace of R^4 spanned by v1 =
[1,1,2,0] and v2 = 2,-1,0,4]. Find a basis for W^T
= {v is in R^2 : w*v = 0 for
w inside of W}

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