givem,-W is the group of all the vectors in R(adsbygoogle = window.adsbygoogle || []).push({}); ^{4}which are orthagonal to the vectors v_{1}(1 0 1 0) and also v_{2}(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

v_{1}(dot)w=0

v_{2}(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

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# Homework Help: Orthagonal vectors/vector spaces

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