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Linearly dependent vectors

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2. Which of the following sets of vectors in R3 are linearly dependent?
a) (4, -1, 2), (-4, 10, 2)
b) (-3, 0, 4), (5, -1, 2), (1, 1, 3)
c) (8, -1, 3), (4, 0, 1)
d) (-2, 0, 1), (3, 2, 5), (6, -1, 1), (7, 0, -2)

4. Which of the following sets of vectors in P2 are linearly dependent?
a) 2 - x + 4x2, 3 + 6x + 2x2, 2 + 10x - 4x2
b) 3 + x + x2, 2 - x + 5x2, 4 - 3x2
c) 6 - x2, 1 + x + 4x2
d) 1 + 3x + 3x2, x + 4x2, 5 + 6x + 3x2, 7 + 2x - x2

12. Show that if {v1, v2, v3} is linearly dependent set of vectors in a vector space V, and v4 is any vector in V, then {v1, v2, v3, v4} is also linear dependent.

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Solution Summary

This set of questions shows how to determine if sets of vectors are linearly dependent.

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For the Linear Independence, the following are important

(1) We call n vectors linearly independent, if implies that .
If you can find which are NOT all zeros so that , then linearly dependent.

(2) Any m n-dimensional vectors are dependent if m>n

General Vector Spaces - Linear Independence

2. Which of the following sets of vectors in R3 are linearly dependent?

a) (4, -1, 2), (-4, 10, 2)

Solution. To see if these two vectors are independent, we let

a(4, -1, 2)+b(-4, 10, 2)=(0,0,0)

Then we get
(4a-4b, -a+10b, 2a+2b)=(0,0,0)
Hence,
4a-4b=0
-a+10b=0
2a+2b=0

So, a=0 and b=0.

Hence, a(4, -1, 2)+b(-4, 10, 2)=(0,0,0) implies that a=0 and b=0.

Hence, (4, -1, 2), (-4, 10, 2) are independent.

b) (-3, 0, 4), (5, -1, 2), (1, 1, 3)

Solution. To see if these vectors ...

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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."
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