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Vectors in 2-Space and 3-Space : Properties of the determinant function

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Please see the attached file for the fully formatted problems.

2. Verify that det(AB) = det(A) det(B) for

A = 2 1 0 and B = 1 -1 3
3 4 0 7 1 2
0 0 2 5 0 1

Is det(A+B) = det(A) + det(B) ?

5. Let A = a b c
d e f
g h i

Assuming that det(A) = -7, find
b) det(A-1)
e) det a g d
b h e
c i f

9. Prove the identity without evaluation the determinants.

a1 + b1 a1 - b1 c1 a1 b1 c1
a2 + b2 a2 - b2 c2 = -2 a2 b2 c2
a3 + b3 a3 - b3 c3 a3 b3 c3

16. Let A and B be n x n matrices. Show that if A is invertible,
then det(B) = det(A-1BA)

18. Prove that a square matrix A is invertible if and only if ATA is invertible.

10. a) In the accompanying figure, the area of the triangle ABC can be expressed as

area ABC = ½ x1 y1 1
x2 y2 1
x3 y3 1

Note: In the derivation of this formula, the vertices are labeled such that the triangle is traced counterclockwise proceeding from (x1, y1) to (x2, y2) to (x3, y3). For a clockwise orientation, the determinant above yields the negative of the area.

b) Use the result in (a) to find the area of the triangle
with vertices (3,3), (4,0), (-2, -1).

Figure Ex-10

Euclidean Vector Spaces: Euclidean n-Space

6. Let u = (4, 1, 2, 3), v = (0, 3, 8, -2), and w = (3, 1, 2, 2). Evaluate each expression.
a) u + v
b) u + v
c) -2u + 2 u
d) 3u - 5v + w
e) 1
w
f) 1
w

16. Find two vectors of norm 1 that are orthogonal to the tree vectors u = (2, 1, -4),
v = (-1, -1, 2, 2), and w = (3, 2, 5, 4).

20. Find u ? v given that u + v = 1 and u - v = 5

24. Prove the following generalization of Theorem 4.1.7. If v1, v2, ..., vr are pairwise orthogonal vectors in Rn, then

v1 + v2 + ... + vr 2 = v1 2 + v2 2 + ... + vr 2

26. Use the Cauchy-Schwarz inequality to prove that for all real values of a, b, and theta,

(a cos(theta) + b sin(theta))^2 =< a^2 + b^2

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Vectors in 2-Space and 3-Space and properties of the determinant function are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
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  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
  • "excellent work"
  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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