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