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Vectors, Basis, Row Space, Column Space and Null Space

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1. Which of the following sets of vectors are bases and why are they bases for P2
A) 1-3x+2x^2, 1+x+4x^2, 1-7x
B) 4+6x+x^2, -1+4x+2x^2, 5+2x-x^2
C) 1+x+x^2, x+x^2, x^2

2. In each part use the information in the table to find the dimension of the row-space, column-space and null-space of A and the null space of AT

Note A = a thru g

a b c d e f g
Size of A 3 x 3 3 x 3 3 x 3 5 x 9 9 x 5 4 x 4 6 x 2

Rank (A) 3 2 1 2 2 0 2

3. Find a basis for the null space of A.

1 -1 3
a) A= 5 -4 -4
7 -6 2

2 0 -1
b) A= 4 0 -2
0 0 0

1 4 5 2
c) A= 2 1 3 0
0.1 3 2 2

1 4 5 6 9
3 -2 1 4 -1
d) A= -1 0 -1 -2 -1
2 3 5 7 8

1 -3 2 2 1
0 3 6 0 -3
e) A= 2 -3 -2 4 4
3 -6 0 6 5
-2 9 2 -4 -5

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1. Which of the following sets of vectors are bases and why are they bases for P2
A) 1-3x+2x^2, 1+x+4x^2, 1-7x

(-2)*(1-3x+2x^2)+1*(1+x+4x^2)+1*(1-7x)=0

So, they are linearly dependent. Therefore, they are NOT bases for P2.

B) 4+6x+x^2, -1+4x+2x^2, 5+2x-x^2

1*(4+6x+x^2)+(-1)*(-1+4x+2x^2)+(-1)*(5+2x-x^2)=0

So, they are linearly dependent. Therefore, they are NOT bases for P2.

C) 1+x+x^2, x+x^2, x^2

If a(1+x+x^2)+b(x+x^2)+cx^2=0, then we have
(a+b+c)x^2+(a+b)x+a=0

Hence,
a+b+c=0
a+b=0
a=0
Hence, a=b=c=0

So, 1+x+x^2, x+x^2, x^2 are linearly independent. Therefore, they are bases for P2.

2. In each part use the information in the table to find the dimension of the row space, column space and null space of A and the null space of AT

Note A = a thru g

a b c d e f g
Size of A 3 x 3 3 x 3 3 x 3 5 x 9 9 x 5 4 x 4 6 x 2

Rank (A) 3 2 1 2 2 0 2

We use the ...

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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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