U a Subspace of R3: Yes- Explained Here

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In summary, U is a subspace of R^3 because it can be expressed as the span of the column vectors [0 1 0]^T and [0 0 1]^T. This can be seen by writing the column vectors as row vectors and noticing that they form a basis for U. For the second question, to determine if U is a subspace of R^3, one would need to check if it satisfies the definition of a subspace. The possible values for r and s that satisfy r^2 + s^2 = 0 are r = 0 and s = 0. The possible vectors in this vector space are {[0 0 0]^T}, as there is only one
  • #1
KataKoniK
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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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  • #2
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.
 
  • #3
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.
 
  • #4
So basically we got to chose arbitrary (sp?) choose a "s" and "t", so that it satisfies the three main conditions?
 
  • #5
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?
 
  • #6
What are the possible (real) values for r and s so that s2+ y2= 0? What are the possible vectors in this vector space?
 
  • #7
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?
 
  • #8
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?
 
  • #9
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?
 

1. What is a subspace?

A subspace is a subset of a vector space that satisfies the three axioms of closure under addition, closure under scalar multiplication, and contains the zero vector. In simpler terms, it is a space that is closed under vector addition and scalar multiplication.

2. How do you know if a subset is a subspace of R3?

To determine if a subset is a subspace of R3, you need to check if it satisfies the three axioms of closure under addition, closure under scalar multiplication, and contains the zero vector. If it satisfies all three axioms, then it is considered a subspace of R3.

3. What is the importance of subspaces in mathematics and science?

Subspaces are important in mathematics and science because they allow us to simplify complex systems and analyze them using vector operations. They also provide a framework for understanding higher-dimensional spaces and solving problems involving linear equations.

4. Can a subspace of R3 be represented geometrically?

Yes, a subspace of R3 can be represented geometrically as a plane passing through the origin. This is because R3 is a three-dimensional space and a subspace of R3 has two dimensions, which can be visualized as a plane.

5. How can subspaces be used in real-world applications?

Subspaces have many real-world applications, such as in computer graphics, data compression, and physics. In computer graphics, subspaces are used to create 3D models, animations, and special effects. In data compression, subspaces are used to reduce the size of large datasets while preserving important information. In physics, subspaces are used to describe the movement of particles and objects in space.

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