In: Advanced Math

Work out the distance of the point [1, 0, 0, 0] from the subspace W = ker T, where T : R 4 → R 2 is the linear map with matrix A = {(1,2,2,0), (2,1,3,1)}

Compute the distance from the point x = (-5,5,2) to the subspace
W of R3 consisting of all vectors
orthogonal to the vector (1, 3, -2).

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

Use Lagrange multipliers to find the distance from the point
(2, 0, −1)
to the plane
4x − 3y +
8z + 1 = 0.

The equation 2x1-3x2=5 defines a line in R2.
a. Find the distance from the point w=(3,1) to
the line by using projection.
b. Find the point on the line closest to w by
using the parametric equation of the line through
w with vector a.

Determine whether or not W is a subspace of R3 where
W consists of all vectors (a,b,c) in R3 such that
1. a =3b
2. a <= b <= c
3. ab=0
4. a+b+c=0
5. b=a2
6. a=2b=3c

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

Determine whether or not W is a subspace of V. Justify your
answer.
W = {p(x) ∈ P(R)|p(1) = −p(−1)}, V =
P(R)

A particle moves from point A = (0, 0, 0) to point B = (2π, 0,
2π), under the action of the force F = xi +
yj − zk .
a. Calculate the work done by the force F on
the particle if it moves along the conic-helical curve
r(t) = (t cost )i +
(t sint )j +
tk with 0 ≤ t ≤ 2π.
b. Find a parametric vector equation for the straight line
connecting A to...

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O
W
L
S
f(O,W,L,S)
0
0
0
0
0
0
1
0
0
0
1
0
2
0
0
1
0
1
3
0
0
1
1
1
4
0
1
0
0
0
5
0
1
0
1
1
6
0
1
1
0
1
7
0
1
1
1
X
8
1
0
0
0
0
9
1
0
0
1
0
10
1
0
1
0
0
11
1
0
1
1
1
12
1...

If V (dimension k-1) is a subspace of W (dimension K),
and V has an orthonormal basis {v1,v2.....vk-1}. Work out a
orthonormal basis of W in terms of that of V and the orthogonal
complement of V in W.
Provide detailed reasoning.

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