Subspace Questions: Checking 2 Sets in R^3

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Homework Help Overview

The discussion revolves around determining whether two specific sets in R^3 are subspaces. The first set consists of all combinations of the vectors (1,1,0) and (2,0,1), while the second set is defined by a plane of vectors satisfying the equation b3 - b2 + 3b1 = 0.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the definition of a subspace, questioning the meaning of "all combinations" and discussing linear combinations of vectors. There is also an inquiry into how to verify the closure properties of the sets under addition and scalar multiplication.

Discussion Status

Participants are actively engaging with the concepts, with some offering definitions and clarifications. There is an exploration of how to approach the verification of the subspace properties for both sets, although no consensus has been reached on the specific methods to apply.

Contextual Notes

One participant expresses uncertainty about the approach to take for the two sets, and there are additional questions regarding operations on matrices and their implications for subspaces.

EvLer
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I am not sure about these 2 whether they are subspaces or not (i do know how to check whether it is a subspace or not)

subsets of R^3, subspace or not?
1.all combinations of (1,1,0) and (2,0,1)
2.plane of vectors (b1,b2,b3) that satisfy b3-b2+3b1 = 0

thanks for help.
 
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What does a subspace mean to you?
 
closed under addition and scalar multiplication... add two vectors and they should still be in the space, multiply by a const and the vector is still in space... also has to contain zero.
i am not sure which side to approach THESE two...

and what in the world is meant by "all combinations"?
 
By combinations, I assume they mean linear combinations. A linear combination of a set of vectors v1,...,vn is any vector of the form a1 v1+...+an vn where the ak are scalars. The set of such vectors is clearly a subspace, and this is sometimes taken as the definition. For the other one, assume two vectors satisfy the equation and show their sum and scalar multiples do as well.
 
StatusX said:
For the other one, assume two vectors satisfy the equation and show their sum and scalar multiples do as well.

scalar multiple I get, but for sum not exactly sure:
(u3+v3) - (u2+v2) + 3(u1+v1) = 0
what does that give me?
 
u3-u2+3u1=0, and similarly for v, so...
 
oh ok, thanks.

one other quick question: if I factor a constant out of a row of a matrix B and get matrix A then can i say that B = 2A?
what if i factor a constant out of a column?
thanks again...

edit: also, how would i show that column spaces of 2 matrices are equal?
 
Last edited:

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