# 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

#### 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.