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.

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

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...

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)

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

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.

a.)
Find the shortest distance from the point (0,1,2) to any point
on the plane x - 2y +z = 2 by finding the function to optimize,
finding its critical points and test for extreme values using the
second derivative test.
b.)
Write the point on the plane whose distance to the point (0,1,2)
is the shortest distance found in part a) above. All the work
necessary to identify this point would be in part a). You just need
to...

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...

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