Homework Answers
1. Let and . Find the eigenvalues of this matrix and determine if it is invertible. In other wo...
1. Let and . Find the eigenvalues of this matrix and determine if it is invertible. In other words, how does finding a basis of for which the matrix of is upper triangular help find the eigenvalues of and how does it help determine is is invertible? 2. Define by . Find all the eigenvalues and eigenvectors of . Note stands for either or . TE L(V) 0 0 8 We were unable to transcribe this imageWe were unable to...
Problem 5. Let E1 = Q(2,7 ), E2= (2,), 1 = 22 + 77, and 2...
Problem 5. Let E1 = Q(2,7
), E2= (2,),
1 = 22
+ 77,
and
2 = 22
+ 3()
(i) Determine [Ei : Q] for i = 1, 2.
(ii) Determine a basis of Ei over Q for i = 1, 2.
(iii) Determine the minimal polynomial of
i over Q for i = 1, 2.
(iv) Determine if each of the extensions E1 / Q and
E2 / Q is Galois.
We were unable to transcribe this imageWe...
2 3 -6 9 0 1 -2 0 3. Let A= 2 -4 7 2 The...
2 3 -6 9 0 1 -2 0 3. Let A= 2 -4 7 2 The RREF of A iso 0 1 3 -6 6 -6 0 0 0 (a) (6 points) Find a basis for Col A, the column space of A. 0 (b) (2 points) What is rank A? (c) (6 points) Find a basis for Null A, the null space of A. (d) (2 points) What is the dimension of the null space of A?
Let S ⊆ be the tetrahedron having vertices (0, 0, 0), (0, 1, 1), (1, 2,...
Let S ⊆
be the tetrahedron having vertices (0, 0, 0), (0, 1, 1), (1, 2,
3), and (−1, 0, 1).
Let f :
→
be the function defined by f(x, y, x) = x − 2y + 3z.
Using the change of variables theorem,
rewrite
as an integral over a 3-rectangle, then use Fubini’s theorem to
evaluate the integral.
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(2 points) Let 4- -1 01 1 1-1 0-2]. Find orthonormal bases of the kernel, row...
(2 points) Let 4- -1 01 1 1-1 0-2]. Find orthonormal bases of the kernel, row space, and image (column space) of A (b) Basis of the row space: (c) Basis of the image (column space)
Let X(t) = 2; if 0 t 1; 3; if 1 t 3; -5; if 3 t 4: or in one formula X(t) = 2I[0;1](t) + 3I...
Let X(t) =
2; if 0 t 1;
3; if 1 t 3;
-5; if 3 t 4:
or in one formula X(t) = 2I[0;1](t) +
3I(1;3](t) -
5I(3;4](t).
Give the Itˆo integral
X(t)dB(t)
as a sum of random variables, give its distribution,
specify the mean and the variance.
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe...
Let X1, X2,.......Xn be a random sample of size n from a continuous distribution symmetric about...
Let X1, X2,.......Xn be a
random sample of size n from a continuous distribution symmetric
about .
For testing H0: =
10 vs H1: <
10, consider the statistic T- =
Ri+ (1-i),
where i
=1 if Xi>10 , 0 otherwise; and
Ri+ is the rank of (Xi - 10) among
|X1 -10|, |X2-10|......|Xn
-10|.
1. Find the null mean and variance of T- .
2. Find the exact null distribution of T- for
n=5.
We were unable to transcribe this imageWe were...
1. Find , where s is , . 2. Find , and , lying inside and underneath 3. Find , where in ...
1. Find , where s is , . 2. Find , and , lying inside and underneath 3. Find , where in cylinder, and between y=0 and y=1 in the first octant. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageds 2az - We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageryzds We were unable to transcribe this image ds...
Given that A = 54 0 LO 3 -2 3 0] 0 has eigenvalues 11 =...
Given that A = 54 0 LO 3 -2 3 0] 0 has eigenvalues 11 = –2 and 12 = 4 and 4] 1 a basis for Exy is 1-2 %. 1] Choose ALL the statement(s) that are ALWAYS TRUE. = -2 are O A is NOT diagonalizable since the algebraic multiplicity and the geometric multiplicity of x different. A is NOT diagonalizable since the algebraic multiplicity and the geometric multiplicity of 12 = 4 are different. O A is...
Let V be a Hilbert space. Let S1 and S2 be two hyperplanes in V defined by Let be given. We con...
Let V be a Hilbert space. Let S1 and S2 be two hyperplanes in V defined by Let be given. We consider the projection of y onto , i.e., the solution of (1) (a) Prove that is a plane, i.e., if , then for any . (b) Prove that z is a solution of (1) if and only if and (2) (c) Find an explicit solution of (1). ( d) Prove the solution found in part (c) is unique. We...
Problem 5. Let E1 = Q(2,7
), E2= (2,),
1 = 22
+ 77,
and
2 = 22
+ 3()
(i) Determine [Ei : Q] for i = 1, 2.
(ii) Determine a basis of Ei over Q for i = 1, 2.
(iii) Determine the minimal polynomial of
i over Q for i = 1, 2.
(iv) Determine if each of the extensions E1 / Q and
E2 / Q is Galois.
We were unable to transcribe this imageWe...
2 3 -6 9 0 1 -2 0 3. Let A= 2 -4 7 2 The RREF of A iso 0 1 3 -6 6 -6 0 0 0 (a) (6 points) Find a basis for Col A, the column space of A. 0 (b) (2 points) What is rank A? (c) (6 points) Find a basis for Null A, the null space of A. (d) (2 points) What is the dimension of the null space of A?
Let S ⊆
be the tetrahedron having vertices (0, 0, 0), (0, 1, 1), (1, 2,
3), and (−1, 0, 1).
Let f :
→
be the function defined by f(x, y, x) = x − 2y + 3z.
Using the change of variables theorem,
rewrite
as an integral over a 3-rectangle, then use Fubini’s theorem to
evaluate the integral.
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image}
(2 points) Let 4- -1 01 1 1-1 0-2]. Find orthonormal bases of the kernel, row space, and image (column space) of A (b) Basis of the row space: (c) Basis of the image (column space)
Let X(t) =
2; if 0 t 1;
3; if 1 t 3;
-5; if 3 t 4:
or in one formula X(t) = 2I[0;1](t) +
3I(1;3](t) -
5I(3;4](t).
Give the Itˆo integral
X(t)dB(t)
as a sum of random variables, give its distribution,
specify the mean and the variance.
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe...
Let X1, X2,.......Xn be a
random sample of size n from a continuous distribution symmetric
about .
For testing H0: =
10 vs H1: <
10, consider the statistic T- =
Ri+ (1-i),
where i
=1 if Xi>10 , 0 otherwise; and
Ri+ is the rank of (Xi - 10) among
|X1 -10|, |X2-10|......|Xn
-10|.
1. Find the null mean and variance of T- .
2. Find the exact null distribution of T- for
n=5.
We were unable to transcribe this imageWe were...
Given that A = 54 0 LO 3 -2 3 0] 0 has eigenvalues 11 = –2 and 12 = 4 and 4] 1 a basis for Exy is 1-2 %. 1] Choose ALL the statement(s) that are ALWAYS TRUE. = -2 are O A is NOT diagonalizable since the algebraic multiplicity and the geometric multiplicity of x different. A is NOT diagonalizable since the algebraic multiplicity and the geometric multiplicity of 12 = 4 are different. O A is...
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