Question

Question #42ca1

Answer 1

See below.

Explanation:

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`