In: Advanced Math

Let W be a subspace of R^n, and P the orthogonal projection onto W. Then Ker P is W^perp.

Problem 4. Let P be the orthogonal
projection associated with a closed subspace S in a
Hilbert space H, that is P is a linear operator
such that
P(f) = f if f ∈ S
and P(f) = 0 if f ∈ S⊥.
(a) Show that P2 = P and
P∗ = P.
(b) Conversely, if P is any bounded operator satisfying
P2 = P and P∗ = P,
prove that P is the orthogonal projection for some closed
subspace...

2a. Find the orthogonal projection of [9,40,-29,4] onto the
subspace of R4 spanned by [1,6,5,6] and [5,1,5,5].
Answer choices: [2,14,-15,7] [-32,13,-10,7] [0,9,12,6]
[-5,-2,3,2] [-12,0,-9,-9] [-16,20,0,4] [27,29,29,21] [-3,1,2,7]
[-23,7,-3,-9] [-15,5,-15,30]
2b. Find the orthogonal projection of [17,18,-10,24] onto the
subspace of R4 spanned by [2,7,1,6] and [3,7,3,4].
Answer choices: [-34,-22,-29,-34] [-6,4,-2,0] [-12,36,21,33]
[3,21,-3,24] [7,-14,-12,1] [5,3,32,45] [14,32,12,11] [9,13,18,11]
[20,2,-3,19] [-2,-6,1,-7]

Using least squares, find the orthogonal projection of
u onto the subspace of R4
spanned by the vectors v1,
v2, and v3,
where
u = (6, 3, 9, 6), v1
= (2, 1, 1, 1), v2 = (1, 0, 1 ,1),
v3 = (-2, -1, 0, -1).

Let V -Φ -> W be linear. Show that ker (Φ) is a
subspace of V and Φ (V) is a subspace of W.

Let W be a subspace of Rn with an orthogonal basis {w1, w2,
..., wp} and let {v1,v2,...,vq} be an orthogonal basis for W⊥.
Let
S = {w1, w2, ..., wp, v1, v2, ..., vq}.
(a) Explain why S is an orthogonal set. (b) Explain why S
spans Rn.
(c) Showthatdim(W)+dim(W⊥)=n.

Let W be a subspace of R^n and suppose that v1,v2,w1,w2,w3 are
vectors in W. Suppose that v1; v2 are linearly independent and that
w1;w2;w3 span W.
(a) If dimW = 3 prove that there is a vector in W that is not
equal to a linear combination of v1 and v2.
(b) If w3 is a linear combination of w1 and w2 prove that w1 and
w2 span W.
(c) If w3 is a linear combination of w1 and...

Let W be a subspace of Rn. Prove that W⊥ is also a
subspace of Rn.

Check the true statements below:
A. The orthogonal projection of y onto v is the same as the
orthogonal projection of y onto cv whenever c≠0.
B. If the columns of an m×n matrix A are orthonormal, then the
linear mapping x→Ax preserves lengths.
C. If a set S={u1,...,up} has the property that ui⋅uj=0 whenever
i≠j, then S is an orthonormal set.
D. Not every orthogonal set in Rn is a linearly independent
set.
E. An orthogonal matrix is invertible.

(a) Find the matrix representation for the orthogonal projection
Pr : R 4 → R 4 onto the plane P= span
1
-1
-1
1
-1
-1
1
1
(b) Find the distance of vector ~y =
2
0
0
4
from the plane P.

If T : R 3 → R 3 is projection onto a line through the origin,
describe geometrically the eigenvalues and eigenvectors of T.

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