Exercise 3.4.12

Answers

1.

Set

v_{1} = (0, - 1, 0, 1, 1, 0)

and

v_{2} = (1, 0, 1, 1, 1, 0)

. Check the two vectors satisfy the system of linear equation and so they are vectors in

V

. To show they are linearly independent, assume that

a (0, - 1, 0, 1, 1, 0) + b (1, 0, 1, 1, 1, 0)

= (b, - a, b, a + b, a + b, 0) = 0 .

This means that $a = b = 0$ and the set is independent.

2.

Similarly we find a basis

β = {(1, 1, 1, 0, 0, 0), (- 1, 1, 0, 1, 0, 0)

, (1, - 2, 0, 0, 1, 0), (- 3, - 2, 0, 0, 0, 1)}

for $V$ as what we do in the Exercise 3.4.4. Still remember that we should put $v_{1}$ and $v_{2}$ on the first and the second column.

(\begin{matrix} 0 & 1 & 1 & - 1 & 1 & - 3 \\ - 1 & 0 & 1 & 1 & - 2 & - 2 \\ 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix})

⇝ (\begin{matrix} 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix})

So the set

{(0, - 1, 0, 1, 1, 0), (1, 0, 1, 1, 1, 0), (- 1, 1, 0, 1, 0, 0), (- 3, - 2, 0, 0, 0, 1)}

forms a basis for $V$ .

jephianlin

2011-06-27 00:00