Exercise 3.4.13

Answers

1.

Set

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

and

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

= (a, 2 b, a + b, a + b, a, 0) = 0 .

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

2.

Take the same basis

β

as that in the previous exercise and do Gaussian elimination.

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

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

So the set

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

forms a basis for $V$ .

jephianlin

2011-06-27 00:00