5 questions in linear algebra

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nynjnq93

Mathematics

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5 clear questions in linear algebra class (most of them multiple parts). The assignment is due 2 days.

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3. (worth 20 points) Find the area of the triangle in 3-space that has the given vertices. P.:(2,6, -1), Pz(1, 1, 1), P3(4,6, 2) 4. (worth 20 points) Consider the following matrices and then compute the given expressions where possible. If not possible, then tell why the given expression is not possible. 30 A = -1 2 1 1 - [1] -- [141] a) 2A+C b) BT+50" c) C-%A d) 8-B e) Tr (BC) f) C(BA) 5. (worth 20 points) For the following matrices, find the characteristic equation, the eigenvalues and the bases for the eigenspaces. a. b. 10 -9 [:] [ -1 4 -2 2. (worth 20 points) A corporation wants to lease a fleet of 12 airplanes with a combined carrying capacity of 220 passengers. The three available types of planes carry 10, 15, and 20 passengers, respectively. Follow the steps below to find out how many of each type of plane should be leased. a. Let x1 = number of 10-passenger planes; Let x2 = number of 15-passenger planes; Let X3 = number of 20-passenger planes. What are the two linear equations that you must solve? b. Write the augmented coefficient matrix from the system of linear equations in part a. c. Solve the system of linear equations by Gauss-Jordan elimination. d. You should result in a solution where the solution to xi and x2 depends on X3. Let x3 = t, where t is any real number. Since you cannot lease fractional planes or negative planes, find the possible values for t. e. Given the possible values of t found in part d, create a table to list the possible solutions for X1, X2, X3. f. If the cost of leasing a 10-passenger plane is $8,000 per month, a 15-passenger plane is $14,000 per month, and a 20-passenger plane is $16,000 per month, which of the three possible solutions in your table from part (2e) would minimize the monthly leasing cost? 1. (worth 20 points) Refer to matrix A below. A = 1 2 3 293 104 a. List the column vectors of A as V1, V2, V3. b. Show that the column vectors of A form a basis for R?. c. Using matrix A above, show that vector b =(5, -1, 9) is in the column space of A. d. Find the coordinate vector of v = (5,-1,9) relative to the basis S = { V1, V2, V3 }, where V1, V2, and V3 are the column vectors from part (a) above.
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Explanation & Answer

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1 2 3
1. 𝐴 = [2 9 3].
1 0 4
𝟏
𝟐
𝟑
a. The column vectors are 𝒗𝟏 = (𝟐), 𝒗𝟐 = (𝟗), 𝒗𝟑 = (𝟑).
𝟏
𝟎
𝟒
b. To prove this it is sufficient to prove that det 𝐴 is nonzero.
1 2 3
det 𝐴 = |2 9 3| = 36 + 6 + 0 − 27 − 0 − 16 = −𝟏 ≠ 0.
1 0 4
c. As we know from (b), any vector is in the column space of 𝐴.
1 2 3 5
d. To do this is the same as to solve the linear system (2 9 3 −1).
1 0 4 9
Find the corresponding determinants:
5 2 3
𝑑1 = |−1 9 3| = 180 + 54 + 0 − 243 − 0 + 8 = −1,
9 0 4
1 5 3
𝑑2 = |2 −1 3| = −4 + 15 + 54 + 3 − 27 − 40 = 1,
1 9 4
1 2 5
𝑑3 = |2 9 −1| = 81 − 2 + 0 − 45 − 0 − 36 = −2.
1 0 9
This way the answer is (𝟏, −𝟏, 𝟐).

2.
a. The number of airplanes is 12, so the first equation is 𝒙𝟏 + 𝒙𝟐 + 𝒙𝟑 = 𝟏𝟐.
The capacity is 220, so the second equation is 10𝑥1 + 15𝑥2 + 20𝑥3 = 220, or
𝟐𝒙𝟏 + 𝟑𝒙𝟐 + 𝟒𝒙𝟑 = 𝟒𝟒.
b. The matrix is (

𝟏
𝟐

𝟏 𝟏
𝟑 𝟒

𝟏𝟐
).
𝟒𝟒

1 1 1 12
).
0 1 2 20
1 0 −1 −8
Then multiply the second row by −1 and add it to the first: (
).
0 1 2 20
c. Multiply the first row by −2 and add it to the second: (

This matrix means 𝒙𝟏 = 𝒙𝟑 − 𝟖, 𝒙𝟐 = −𝟐𝒙𝟑 + 𝟐𝟎. It is the general solution (𝑥3 may be
arbitrary).
d. As we see from the first equation, 𝑥1 = 𝑡 − 8, 𝑡 must be an integer not less than 8.
As we see from the second equation, 𝑥2 = 20 − 2𝑡, 𝑡 must be not greater than ...


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