- #1

Dell

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^{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 for

*W+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), doesn't 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)??