# Thread: Spanning sets

1. ## Spanning sets

Determine if each of the following are true or false, and give a justification.

(a) Let v1, v2 and v3 be vectors in a subspace in V. If { v1, v2} is a spanning set of V, then {v1, v2,v3} must be a spanning set of V.

On this one I put true because v3 could be a linear combination of v1, v2.

(b) Let v1, v2,v3 be vectors in subspace V. If {v1, v2,v3} is a spanning set of V, then { v1, v2} must be a spanning set of V.

Here I put false because if {v1, v2,v3} is linearly dependent, then a vector can't be removed from the set.

(c) Every spanning set of R2 must contain at least two vectors.

Here I put false but I don't really know why.. I know if I have an image of a matrix [-s,s]^T then the spanning set would only be a single vector [-1,1]^T...

Am I on the right track for any of these?

2. ## Re: Spanning sets Originally Posted by MrJank Determine if each of the following are true or false, and give a justification.
(a) Let v1, v2 and v3 be vectors in a subspace in V. If { v1, v2} is a spanning set of V, then {v1, v2,v3} must be a spanning set of V.
On this one I put true because v3 could be a linear combination of v1, v2.
(b) Let v1, v2,v3 be vectors in subspace V. If {v1, v2,v3} is a spanning set of V, then { v1, v2} must be a spanning set of V.
Here I put false because if {v1, v2,v3} is linearly dependent, then a vector can't be removed from the set.
(c) Every spanning set of R2 must contain at least two vectors.
Here I put false but I don't really know why.. I know if I have an image of a matrix [-s,s]^T then the spanning set would only be a single vector [-1,1]^T...Am I on the right track for any of these?
In part (a) do you mean "in a subspace V?" If so, you are correct.

In part (c) what do you think the 2 means in $\mathbb{R}^2~?$

3. ## Re: Spanning sets

I thought R2 was the set of vectors in a 2 dimensional space or all 2x1 vectors.

4. ## Re: Spanning sets Originally Posted by MrJank I thought R2 was the set of vectors in a 2 dimensional space or all 2x1 vectors.
Is it possible for one $2\times 1$ vector to generate all vectors in $\mathbb{R}^2~?$

5. ## Re: Spanning sets

When you put it that way, no. So is the spanning set like all the vectors needed to have a linear combination of all the vectors in the given subspace?

6. ## Re: Spanning sets Originally Posted by MrJank When you put it that way, no. So is the spanning set like all the vectors needed to have a linear combination of all the vectors in the given subspace?
Then how can part (c) be false?