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

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 ...

Solution Summary

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