Question #42ca1

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Answer 1 · 2016-10-02T11:25:38

See below.

Calling

$X = (x_1,x_2,x_3,x_4)$
$S->((s_1),(s_2))=((1,2,1,-1),(0,-1,3,1))$ and
$T-> << t_0, X >> =0$ with $t_0=(1,1,0,0)$

The space $(S_|_)nn T$ is

$<< s_1, X_a >> = 0$
$<< s_2,X_a >> = 0$
$<< t_0,X_a >> = 0$

This gives $X_a = lambda (-1,1,0,1), lambda in RR$ so

$W$ the complement, orthogonal to $X_a$ . This space is given by
$X_w | << lambda(-1,1,0,1), X_w >> = 0$
This space is then

$W = << w_0, X >> = -x_1+x_2+x_4=0$

with $w_0 = (-1,1,0,1)$

As can be checked

$det((s_1),(s_2),(t_0),(w_0)) = 12$

so ${s_1,s_2,t_0,w_0}$ span $RR^4$