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in this question i am given 2 subspaces of R

W={(x

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

and am asked to find

1] a basis and dimention for W

2] a basis for W+U

3] a basis for W[tex]\cap[/tex]U

1]

since i only have limitations on x

therefore W=sp{(1,0,0,1),(0,1,0,0),(0,0,1,0)} and dimW=3

2]

to find a basis for W+U, i look for linearly independant vectors that span the space, so i set up a matrix to see which are combinations of the others,

i got W+U=sp{(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)}

these 4 span the whole of R4 so does this mean that W+U = R4

3]

to find a basis of W[tex]\cap[/tex]U, i found a homogenic system for each subspace and compared the 2.

taking a random vector (a b c d) in the subspace, i get

for W==> -4a+b+3c-sd=0

for U==> a-d=0

from the combination i get -6a+b+3c=0

a=t b=6t-3s c=s d=t

and so i get W[tex]\cap[/tex]U=sp{(1,6,0,1)(0,-3,1,0)} dim=2

----------------------------------------

does this all look okay,

also, if i am asked to find a basis and i write the span, is it the same thing, or must i write just one possible basis in the span

^{4}W={(x

_{1},x_{2},x_{3},x_{4})[tex]\in[/tex]R^{4}|x_{1}-x_{4}=0}U=sp{(1,2,0,-1),(2,3,1,-1),(1,-1,1,-1)}

and am asked to find

1] a basis and dimention for W

2] a basis for W+U

3] a basis for W[tex]\cap[/tex]U

1]

since i only have limitations on x

_{1}and x_{4}i call x_{2}=t x_{3}=s x_{1}=x_{4}=qtherefore W=sp{(1,0,0,1),(0,1,0,0),(0,0,1,0)} and dimW=3

2]

to find a basis for W+U, i look for linearly independant vectors that span the space, so i set up a matrix to see which are combinations of the others,

i got W+U=sp{(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)}

these 4 span the whole of R4 so does this mean that W+U = R4

3]

to find a basis of W[tex]\cap[/tex]U, i found a homogenic system for each subspace and compared the 2.

taking a random vector (a b c d) in the subspace, i get

for W==> -4a+b+3c-sd=0

for U==> a-d=0

from the combination i get -6a+b+3c=0

a=t b=6t-3s c=s d=t

and so i get W[tex]\cap[/tex]U=sp{(1,6,0,1)(0,-3,1,0)} dim=2

----------------------------------------

does this all look okay,

also, if i am asked to find a basis and i write the span, is it the same thing, or must i write just one possible basis in the span

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