Orthagonal vectors/vector spaces

[Dell]

givem,-W is the group of all the vectors in R⁴ which are orthagonal to the vectors v₁(1 0 1 0) and also v₂(1 2 1 1)
a) find a basis and the dimension of W
b) find, in W, a vector which is orthagonal to the vector (1 -2 1 3)

a)

i call this vector w=(a b c d) and say
v₁(dot)w=0
v₂(dot)w=0

therefore

a+0b+c+0d=0 a=-c
a+2b+c+d=0 d=-2b

i will have 2 free parameters here, 4 unknows -rank2=2
a=t
d=u

so w=(t, -2u, -t, u)

W=sp{(1 0 -1 0), (0 -2 0 1)}
dim(w)=2
----------------------------
b)

(t, -2u, -t, u) dot (1 -2 1 3)=0

t+4u-t+3u=0
7u=0
u=0

therefore any vector that fits
(t 0 -t 0) is orthagonal to (1 -2 1 3) and a part of W


is this all correct

 

[Mark44]

Looks good.