Question

Consider W = Span{p1(t),p2(t)} where p1(t) = 1−t^2 and p2(t) = 3+2t are the polynomials deﬁned on the interval [−1,1]. Find the orthogonal projection of q(t) = t^2−t−1 onto W.

Answer #1

Consider V = fp(t) 2 P2 : p0(1) = p(0)g, where P2 is a set
of
all polynomials of degree less than or equal to 2.
(1) Show that V is a subspace of P2
(2) Find a basis of V and the dimension of V

A): compute projw j if u1=[-7,1,4] u2=[-1,1,-2],w=span{u1,u2}.
(u1 and u2 are orthogonal)
B): let u1=[1,1,1], u2=1/3 *[1,1,-2] and w=span{u1,u2}.
Construct an orthonormal basis for w.

✓For Pi as defined below, show that {P1, P2, P3} is an
orthogonal subset of R4. Find a fourth vector P4 such that {P1, P2,
P3, P4} forms an orthogonal basis in R4. To what extent is P4
unique?
P1 = [1,1,1,1]t, P2 = [1, −2, 1, 0]t, P3 = [1,1,1, −3]t

Determine if the given polynomials given below span
P2
p(x) = 1 − x2 , q(x) = 1 + x, r(x) = 4x2+
3x − 1, s(x) = 3x2+ 4x + 1

P(W < t) = 1/12(t^2+ 2t^3) for 0 ≤ t ≤ 2. Write an integral
for E[√(W+ 3) ].

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

2. Let P 1 and P2 be planes with general equations P1 : −2x + y
− 4z = 2, P2 : x + 2y = 7.
(a) Let P3 be a plane which is orthogonal to both P1 and P2. If
such a plane P3 exists, give a possible general equation for it.
Otherwise, explain why it is not possible to find such a plane. (b)
Let ` be a line which is orthogonal to both P1 and P2....

The projection of the vector (1,1,-1) onto the subspace
span{(1,-1,1), (1,1,0)} is given by:
a)(1,-1,0)
b)(1,1,0)
c) (-1/3, 1)
d)(2, 4, -1)
e)(2/3, 4/3, -1/3)
f)None of the above

Consider the transformation, T : P1 → P2 defined by T(ax + b) =
ax2 + ax + a
(a) Find the image of 2x + 1.
(b) Find another element of P1 that has the same image.
(c) Is T a one-to-one transformation? Why or why not?
(d) Find ker(T) and determine the basis for ker(T). What is the
dimension of kernel(T)?
(e) Find range(T) and determine a basis for range(T). What is
the dimension of range(T)?

1. Consider this hypothesis test:
H0: p1 - p2 = 0
Ha: p1 - p2 > 0
Here p1 is the population proportion of “yes” of
Population 1 and p2 is the population proportion of
“yes” of Population 2. Use the statistics data from a simple random
sample of each of the two populations to complete the following:
(8 points)
Population 1
Population 2
Sample Size (n)
400
600
Number of “yes”
300
426
Compute the test statistic z.
What...

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