i am given 2 subspaces of R(adsbygoogle = window.adsbygoogle || []).push({}); ^{4}

W=sp{(a-b,a+2b,a,b)|a,b[tex]\in[/tex]R}

U=sp{(1,0,1,1)(-6,8,-3,-2)}

and am asked to find:

a homogenic system for W- system for a vector (x,y,z,t) belonging to W

i see the basis for W is :a(1,1,1,0)+b(-1,2,0,1),, i put these vectors into an extended matrix with (x y z t) on the other side, and after a series of elementary operations, i get

x+t-z=0

y-z-2t=0

next i am asked to find

a basis for W+U and W[tex]\cap[/tex]U

forW[tex]\cap[/tex]U

i find a homogenic system for U, and compre it with the system i found for W which comes to

x+t-z=0

y-z-2t=0

y+8z-8t=0

3t-4z+x=0

and i come to

t=1.5z

z=z

y=4z

x=-0.5z

so for W[tex]\cap[/tex]U i get a basis (-0.5, 4, 1, 1.5)

for W+U i take the basis of each and check independace of all of them together, in which i get that all4 are independant, therefore the basis forW+U={(1,0,1,1)(-6,8,-3,-2)(1,1,1,0)(-1,2,0,1),}

if i perform elementary colum operations on them i can get to (1000)(0100)(0010)(0001), doesnt this mean that W+U is the whole vector space R^{4}??

the final question is

find a vector other than the zero vector which is orthagonal to all the vectors in U+W

is this possible, since i found that W+U is the whole vector space R^{4}(supposing i was correct there)??

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# Homework Help: Vector space

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