What's the difference between and , and are vector spaces. I didn't find the answer on the Internet.
A friend told me that . But then if and are as vector space over then I get that is formed by ... ahh well I'm unsure of everything.
What's the difference between and , and are vector spaces. I didn't find the answer on the Internet.
A friend told me that . But then if and are as vector space over then I get that is formed by ... ahh well I'm unsure of everything.
Well, yes, and since is already a subspace its span is also just . That's not a very good example!
Suppose V= {(x, 0, 0)} in and W= {(0, y, 0)} in . Then is the set of all vectors of the form (x, 0, 0) or (0, y, 0). Geometrically, that is the x-axis and the y-axis but NOT the points in between. That is not a subspace because, while it contains (1, 0, 0) and (0, 1, 0), it does NOT contain their sum, (1, 1, 0). The direct sum of V and W is the xy-plane: {(x, y, 0)}} which includes (1, 0, 0), (0, 1, 0) and (1, 1, 0).