# Thread: Sum of Linear Spaces

1. ## Sum of Linear Spaces

Hi. Glad the site is running again. By the way, where is the latex add-on?

Can someone please explain the differences between

1. PQ, P and Q are linear spaces

2. P + Q

3. Sp(PQ)

Are 2 and 3 the same?

2. P union Q will equal the set of all vectors coming from either P or Q.

P + Q will equal the set of all vectors that look like p + q, where p in P and q in Q.

Sp ( P union Q ) will equal the set of all linear combinations of vectors coming from P union Q.

1 and 2 are not necessarily the same, I wouldn't think: the zero vector of one space need not be the zero vector of another space, although the zero vector of any given space is unique.

That's all I have time for right now.

3. well i would think that 2 and 3 are not the same as P + Q is not all linear combinations as it does not include scalar multiplication, where as 3 does?

4. Originally Posted by hmmmm
well i would think that 2 and 3 are not the same as P + Q is not all linear combinations as it does not include scalar multiplication, where as 3 does?
Except that the underlying spaces are linear, which does include scalar multiplication.

5. ah sorry about that you are completely correct i apologise for that, really was not thinking very thourghly