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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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.