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Orthagonal vectors/vector spaces

  • Thread starter Dell
  • Start date
  • #1
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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 dimention 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
 

Answers and Replies

  • #2
33,636
5,296
Looks good.
 

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