Sum of two subspaces - question.

[peripatein]

Homework Statement

Is it possible to add the following subspaces: W_1 = Sp{(1,0,0)} and W_3 = Sp{(0,1,-1), (0,0,1)}?

Homework Equations

The Attempt at a Solution

Will their sum be: Sp{(1,1,-1),(1,0,1)}?

 

[Mark44]

Homework Statement

Is it possible to add the following subspaces: W_1 = Sp{(1,0,0)} and W_3 = Sp{(0,1,-1), (0,0,1)}?

Homework Equations

The Attempt at a Solution

Will their sum be: Sp{(1,1,-1),(1,0,1)}?

No.

It would help to think about the geometry here. What does the W₁ subspace look like? Can you describe it geometrically?

What does the W₃ subspace look like? Can you describe it geometrically?

Are any vectors in W₁ also in W₃? Are any vectors in W₃ also in W₁?

The "sum" of the two sets (actually the direct sum, W₁[itex] \oplus[/itex] W₃) is the set of vectors v such that v = w₁ + w₃, where w₁ ##\in## W₁ and w₃ ##\in## W₃.

 

[peripatein]

I understand that their sum is a direct sum as the intersection is null, but I am not sure how that helps me to find Sp(W_1+W_3).

 

[peripatein]

I was also asked to prove that (W_1 intersection W_2) + (W_1 intersection W_3) = W_1 intersection (W_2+W_3), where W_1, W_2, W_3 are subspaces in vector space V.

Attempt at solution:
Let W_1=Sp(K), W_2=Sp(U), W_3=Sp(L)
Hence, (W_1 intersection W_2) = Sp(K) intersection Sp(U) = Sp(K intersection U)
Hence, (W_1 intersection W_3) = Sp(K) intersection Sp(L) = Sp(K intersection L)
Hence, (W_1 intersection W_2) + (W_1 intersection W_3) = Sp(K intersection U) + Sp(K intersection L) = Sp(K intersection U + K intersection L) = Sp(K intersection (U+L))
I am really not sure this is correct. Is it? Is this really how I should try proving it?

 

[peripatein]

Mark44, their geometrical representation will be:
W_3 = y-(-z) plane and z axis, W_1 = x axis

 

[Michael Redei]

Mark44, their geometrical representation will be:
W_3 = y-(-z) plane and z axis, W_1 = x axis

So W₁ contains all vectors of the form (x,0,0), and W₃ contains all vectors of the form (0,y,z).

What vectors can be expressed as a sum of one vector from W₁ and one from W₃?

 

[peripatein]

R^3.
How may I check whether W_1+W_2=V, where V is a vector space and W_1 and W_2 are two subspaces in V, and W_1=(t,s,t-2s,2t) where t,s belong to R, and W_2=(x,y,z,-x-y-z)?
I first tried to write W_1 as {(1,0,1,2), (0,1,-2,0)} and W_2 as {(1,0,0,-1), (0,1,0,-1),(0,0,1,-1)} and then concluded that since dim(W_1+W_2) was not equal to dim(V) they could not possibly be equal. But I am really not sure that's correct. Could anyone please advise?

 

[Dick]

R^3.
How may I check whether W_1+W_2=V, where V is a vector space and W_1 and W_2 are two subspaces in V, and W_1=(t,s,t-2s,2t) where t,s belong to R, and W_2=(x,y,z,-x-y-z)?
I first tried to write W_1 as {(1,0,1,2), (0,1,-2,0)} and W_2 as {(1,0,0,-1), (0,1,0,-1),(0,0,1,-1)} and then concluded that since dim(W_1+W_2) was not equal to dim(V) they could not possibly be equal. But I am really not sure that's correct. Could anyone please advise?

There's a systematic way to do these problems. Set up a matrix where the rows are your vectors and row reduce it. When you are done the remaining nonzero vectors will be a basis for W_1+W_2.