Subspaces R^n

KataKoniK

Q: Determine whether U is a subspace of R^3.

U = {[0 s t]^T | s and t in R}

A: Yes. U = span {[0 1 0]^T, [0 0 1]}

Can someone explain to me how the heck they come up with that answer? Seems so random.

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Muzza

Let's write column vectors as row vectors, shall we.

U = { (0, s, t); s and t in R } = { (0, s, 0) + (0, 0, t); s and t in R } = { s(0, 1, 0) + t(0, 0, 1); s and t in R }.

See now how one might notice that (0, 1, 0) and (0, 0, 1) form a basis for U?

Of course, this is not necessary to show that U is a subspace of R^3, one can directly use the definition of subspace instead.

matt grime

Homework Helper
not random at all - they are just noting that given two (or any set of) vectors, they span a subspace and the things they span in this case is exactly the set U. it is easy to check the axioms if you need to.

KataKoniK

So basically we got to chose arbitrary (sp?) choose a "s" and "t", so that it satisfies the three main conditions?

KataKoniK

Actually, how would you do this question?

U = {[r 0 s]^T | r^2 + s^2 = 0, r and s in R}

Determine if U is a subspace of R^3?

HallsofIvy

Homework Helper
What are the possible (real) values for r and s so that s2+ y2= 0? What are the possible vectors in this vector space?

KataKoniK

HallsofIvy said:
What are the possible (real) values for r and s so that s2+ y2= 0? What are the possible vectors in this vector space?
r = 0 and s = 0 correct? Possible vectors? That's what I cannot figure out. Are there many different answers for this?

Hurkyl

Staff Emeritus
Gold Member
Well, there's one element in {[r 0 s]^T | r^2 + s^2 = 0, r and s in R} for each r,s pair satisfying r^2 + s^2 = 0, right?

KataKoniK

So from what you are saying then yes, there is an element in {[r 0 s]^T | r^2 + s^2 = 0, r and s in R} for each r, s pair satisfying r^2 + s^2 = 0 where r and s = 0?

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